# Cubes and Cube Roots - Class 8 - Mathematics

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## Exercise 4.1 - Cubes and Cube Roots | R.D. Sharma | Mathematics | Class 8

Find the cubes of the following numbers:

(i) 7

(ii) 12

(iii) 16

(iv) 21

(v) 40

(vi) 55

(vii) 100

(viii) 302

(ix) 301

Here are the cubes of the requested numbers:

(i) $7^3 = 7 \times 7 \times 7 = 343$

(ii) $12^3 = 12 \times 12 \times 12 = 1728$

(iii) $16^3 = 16 \times 16 \times 16 = 4096$

(iv) $21^3 = 21 \times 21 \times 21 = 9261$

(v) $40^3 = 40 \times 40 \times 40 = 64000$

(vi) $55^3 = 55 \times 55 \times 55 = 166375$

(vii) $100^3 = 100 \times 100 \times 100 = 1,000,000$

(viii) $302^3 = 302 \times 302 \times 302 = 27,543,608$

(ix) $301^3 = 301 \times 301 \times 301 = 27,270,901$

Write the cubes of all natural numbers between 1 and 10 and verify the following statements:

(i) Cubes of all odd natural numbers are odd.

(ii) Cubes of all even natural numbers are even.

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Sign up nowObserve the following pattern:

$$ \begin{aligned} 1^{3} & =1 \\ 1^{3}+2^{3} & =(1+2)^{2} \\ 1^{3}+2^{3}+3^{3} & =(1+2+3)^{2} \end{aligned} $$

Write the next three rows and calculate the value of $1^{3}+2^{3}+3^{3}+\ldots+9^{3}+10^{3}$ by the above pattern.

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Sign up nowWrite the cubes of 5 natural numbers which are multiples of 3 and verify the followings:

The cube of a natural number which is a multiple of 3 is a multiple of 27

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Sign up nowWrite the cubes of 5 natural numbers which are of the form $3 n+1$ (e.g. $4,7,10, \ldots)$ and verify the following:

The cube of a natural number of the form $3 n+1$ is a natural number of the same form i.e. when divided by 3 it leaves the remainder 1

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Write the cubes of 5 natural numbers of the form $3 n+2$ (i.e. $5,8,11, \ldots)$ and verify the following:

The cube of a natural number of the form $3 n+2$ is a natural number of the same form i.e. when it is dividend by 3 the remainder is 2

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Write the cubes of 5 natural numbers of which are multiples of 7 and verify the following:

The cube of a multiple of 7 is a multiple of $7^{3}$.

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Which of the following are perfect cubes?

(i) 64

(ii) 216

(iii) 243

(iv) 1000

(v) 1728

(vi) 3087

(vii) 4608

(viii) 106480

(ix) 166375

(x) 456533

First find the prime factors of each part and then explain why or why not a number isn't a perfect cube.

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Which of the following are cubes of even natural numbers?

$$216,512,729,1000,3375,13824$$

List down the cube root of each number and then choose the even natural numbers.

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Which of the following are cubes of odd natural numbers?

$$125,343,1728,4096,32768,6859$$

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What is the smallest number by which the following numbers must be multiplied, so that the products are perfect cubes?

(i) 675

(ii) 1323

(iii) 2560

(iv) 7803

(v) 107811

(vi) 35721

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By which smallest number must the following numbers be divided so that the quotient is a perfect cube?

(i) 675

(ii) 8640

(iii) 1600

(iv) 8788

(v) 7803

(vi) 107811

(vii) 35721

(viii) 243000

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Prove that if a number is trebled then its cube is 27 times the cube of the given number.

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What happens to the cube of a number if the number is multiplied by

(i) 3 ?

(ii) 4 ?

(iii) 5 ?

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Find the volume of a cube, one face of which has an area of $64 \mathrm{~m}^{2}$.

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Find the volume of a cube whose surface area is $384 \mathrm{~m}^{2}$.

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Evaluate the following:

(i) $\left\{\left(5^{2}+12^{2}\right)^{1 / 2}\right\}^{3}$

(ii) $\left\{\left(6^{2}+8^{2}\right)^{1 / 2}\right\}^{3}$

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Write the units digit of the cube of each of the following numbers:

$31,109,388,833,4276,5922,77774,44447,125125125$

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Find the cubes of the following numbers by column method:

(i) 35

(ii) 56

(iii) 72

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Which of the following numbers are not perfect cubes?

(i) 64

(ii) 216

(iii) 243

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For each of the non-perfect cubes in Q. No. 20 find the smallest number by which 3 must be

(a) multiplied so that the product is a perfect cube.

(b) divided so that the quotient is a perfect cube.

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By taking three different values of $n$ verify the truth of the following statements:

(i) If $n$ is even, then $n^{3}$ is also even.

(ii) if $n$ is odd, then $n^{3}$ is also odd.

(iii) If $n$ leaves remainder 1 when divided by 3 , then $n^{3}$ also leaves 1 as remainter when divided by 3 .

(iv) If a natural number $n$ is of the form $3 p+2$ then $n^{3}$ also a number of the same type.

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Write true (T) or false (F) for the following statements:

(i) 392 is a perfect cube.

(ii) 8640 is not a perfect cube.

(iii) No cube can end with exactly two zeros.

(iv) There is no perfect cube which ends in 4

(v) For an integer $a, a^{3}$ is always greater than $a^{2}$

(vi) If $a$ and $b$ are integers such that $a^{2}>b^{2}$, then $a^{3}>b^{3}$.

(vii) If $a$ divides $b$, then $a^{3}$ divides $b^{3}$.

(viii) If $a^{2}$ ends in 9 , then $a^{3}$ ends in 7 .

(ix) If $a^{2}$ ends in 5 , then $a^{3}$ ends in 25.

(x) If $a^{2}$ ends in an even number of zeros, then $a^{3}$ ends in an odd number of reros.

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## Exercise 4.2 - Cubes and Cube Roots | R.D. Sharma | Mathematics | Class 8

Find the cubes of:

(i) -11

(ii) -12

(iii) -21

The cubes of the given numbers are as follows:

(i) The cube of $-11$ is $-1331 ((-11) \times (-11) \times (-11))$.

(ii) The cube of $-12$ is $-1728 ((-12) \times (-12) \times (-12))$.

(iii) The cube of $-21$ is $-9261 ((-21) \times (-21) \times (-21))$.

Which of the following numbers are cubes of negative integers

(i) -64

(ii) -1056

(iii) -2197

(iv) -2744

(v) -42875

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Show that the following integers are cubes of negative integers. Also, find the integer whose cube is the given integer.

(i) -5832

(ii) -2744000

Explain with steps.

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Find the cube of:

(i) $\frac{7}{9}$

(ii) $-\frac{8}{11}$

(iii) $\frac{12}{7}$

(iv) $-\frac{13}{8}$

(v) $2 \frac{2}{5}$

(vi) $3 \frac{1}{4}$

(vii) 0.3

(viii) 1.5

(ix) 0.08

(x) 2.1

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Find which of the following numbers are cubes of rational numbers:

(i) $\frac{27}{64}$

(ii) $\frac{125}{128}$

(iii) 0.001331

(iv) 0.04

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## Exercise 4.3 - Cubes and Cube Roots | R.D. Sharma | Mathematics | Class 8

Find the cube roots of the following numbers by successive subtraction of numbers:

$$ 1,7,19,37,61,91,127,169,217,271,331,397, \ldots $$

(i) 64

(ii) 512

(iii) 1728

To find the cube root of a given number by successive subtraction of numbers from the series $1,7,19,37,61,91,127,169,217,271,331,397, \ldots$, we notice that this series represents the difference between consecutive cube numbers starting from $1^3=1$.

The pattern between the numbers in the series is that they are the differences between consecutive cubes, i.e., $(n+1)^3 - n^3$ for $n=1,2,3,\ldots$.

The algorithm to find the cube root by successive subtraction is as follows:

Start with the given number.

Subtract numbers from the provided series starting from the beginning until you cannot subtract without going negative.

The number of times you successfully subtract is the cube root.

Let's apply this algorithm to each of the given numbers:

### (i) 64

We'll subtract numbers from the series $1,7,19,37,61,\ldots$ from $64$ and count the number of subtractions until we can't subtract without going negative.

$$ 64 -1 = 63 $$

$$ 63 - 7 = 56 $$

$$ 56 - 19 = 37 $$

$$ 37 - 37 = 0 $$

Since we had to subtract 4 times, the cube root of $64$ is $4$

### (ii) 512

Similarly, we'll subtract numbers from the series $1,7,19,37,61,\ldots$ from $512$ and count the subtractions.

$$ 512 - 1 = 511 $$

$$ 511 - 7 = 504 $$

$$ 504 - 19 = 485 $$

$$ 485 - 37 = 448 $$

$$ 448 - 61 = 387 $$

$$ 387 - 91 = 296 $$

$$ 296 - 127 = 169 $$

$$ 169 - 169 = 0 $$

Since we had to subtract 8 times, the cube root of $512$ is $8$

### (iii) 1728

And we'll do the same for $1728$.

$$ 1728 - 1 = 1727 $$

$$ 1727 - 7 = 1720 $$

$$ 1720 - 19 = 1701 $$

$$ 1701 - 37 = 1664 $$

$$ 1664 - 61 = 1603 $$

$$ 1603 - 91 = 1512 $$

$$ 1512 - 127 = 1385 $$

$$ 1385 - 169 = 1216 $$

$$ 1216 - 217 = 999 $$

$$ 999 - 271 = 728 $$

$$ 728 - 331 = 397 $$

$$ 397 - 397 = 0 $$

Since we had to subtract 12 times, the cube root of $1728$ is $12$

Using the method of successive subtraction examine whether or not the following numbers are perfect cubes:

(i) 130

(ii) 345

(iii) 792

(iv) 1331

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Find the smallest number that must be subtracted from those of the numbers in question 2 which are not perfect cubes, to make them perfect cubes. What are the corresponding cube roots?

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Find the cube root of each of the following natural numbers:

(i) 343

(ii) 2744

(iii) 4913

(iv) 1728

(v) 35937

(vi) 17576

(vii) 134217728

(viii) 48228544

(ix) 74088000

(x) 157464

(xi) 1157625

(xii) 33698267

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Multiply 210125 by the smallest number so that the product is a perfect cube. Also find out the cube root of the product.

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What is the smallest number by which 8192 must be divided so that quotient is perfect cube? Also, find the cube root of the quotient so obtained.

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Three numbers are in the ratio $1: 2: 3$. The sum of their cubes is 98784 . Find it numbers.

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The volume of a cube is $9261000 \mathrm{~m}^{3}$. Find the side of the cube. Explain with steps.

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## Exercise 4.4 - Cubes and Cube Roots | R.D. Sharma | Mathematics | Class 8

Find the cube roots of each of the following integers:

(i) -125

(ii) -5832

(iii) -2744000

(iv) -753571

(v) -32768

**For $-125$**:The cube root of $-125$ is $-5$.

Resolution into prime factors: $-125 = -1 \times 5 \times 5 \times 5$.

Thus, $\sqrt[3]{-125} = -\sqrt[3]{5^3} = -5$.

**For $-5832$**:The cube root of $-5832$ is $-18$.

Resolution into prime factors: $-5832 = -1 \times 2^3 \times 3^3 \times 3^3$.

Hence, $\sqrt[3]{-5832} = -\sqrt[3]{2^3 \times 3^3} = -18$.

**For $-2744000$**:The cube root of $-2744000$ is $-140$.

Resolution into prime factors: $-2744000 = -1 \times 2^5 \times 5^3 \times 7^3$.

Therefore, $\sqrt[3]{-2744000} = -\sqrt[3]{2^5 \times 5^3 \times 7^3} = -2^{\frac{5}{3}} \times 5 \times 7 = -140$.

**For $-753571$**:The cube root of $-753571$ is $-91$.

The factorization into primes and further explanation for exact decomposition for $-753571$ might require specific computational help, as its prime factorization does not result in a perfect cube. Hence, the cube root results in $-91$ indicating it does not resolve into a simple prime factorization like the others.

**For $-32768$**:The cube root of $-32768$ is $-32$.

Resolution into prime factors: $-32768 = -1 \times 2^{15}$.

Consequently, $\sqrt[3]{-32768} = -\sqrt[3]{2^{15}} = -2^{15/3} = -32$.

Show that:

(i) $\sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{27 \times 64}$

(ii) $\sqrt[3]{64 \times 729}=\sqrt[3]{64} \times \sqrt[3]{729}$

(iii) $\sqrt[3]{-125 \times 216}=\sqrt[3]{-125} \times \sqrt[3]{216}$

(iv) $\sqrt[3]{-125 \times-1000}=\sqrt[3]{-125} \times \sqrt[3]{-1000}$

Explain each part with steps.

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Find the cube root of each of the following numbers:

(i) $8 \times 125$

(ii) $-1728 \times 216$

(iii) $-27 \times 2744$

(iv) $-729 \times-15625$

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Evaluate:

(i) $\sqrt[3]{4^{3} \times 6^{3}}$

(ii) $\sqrt[3]{8 \times 17 \times 17 \times 17}$

(iii) $\sqrt[4]{700 \times 2 \times 49 \times 5}$

(iv) $125 \sqrt[3]{a^{6}}-\sqrt[3]{125 a}$

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Find the cube root of each of the following rational numbers:

(i) $\frac{-125}{729}$

(ii) $\frac{10648}{12167}$

(iii) $\frac{-19683}{24389}$

(iv) $\frac{686}{-3456}$

(v) $\frac{-39304}{-42875}$

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Find the cube root of each of the following rational numbers:

(i) 0.001728

(ii) 0.003375

(iii) 0.001

(iv) 1.331

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Evaluate each of the following:

(i) $\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}$

(ii) $\sqrt[3]{1000}+\sqrt[3]{0.008}-\sqrt[3]{0.125}$

(iii) $\sqrt[3]{\frac{729}{216}} \times \frac{6}{9}$

(iv) $\sqrt[3]{\frac{0.027}{0.008}} \div \sqrt{\frac{0.09}{0.04}}-1$

(v) $\sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13}$

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Show that:

(i) $\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\sqrt[3]{\frac{729}{1000}}$

(ii) $\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\sqrt[3]{\frac{-512}{343}}$

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Fill in the blanks, with steps:

(i) $\sqrt[3]{125 \times 27}=3 \times \ldots$

(ii) $\sqrt[3]{8 \times \ldots}=8$

(iii) $\sqrt[3]{1728}=4 \times \ldots$

(iv) $\sqrt[3]{480}=\sqrt[3]{3} \times 2 \times \sqrt[3]{\ldots}$

(v) $\sqrt[3]{\cdots}=\sqrt[3]{7} \times \sqrt[3]{8}$

(vi) $\sqrt[3]{\ldots}=\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6}$

(vii) $\sqrt[3]{\frac{27}{125}}=\frac{\ldots}{5}$

(viii) $\sqrt[3]{\frac{729}{1331}}=\frac{9}{\ldots}$

(ix) $\sqrt[3]{\frac{512}{\ldots}}=\frac{8}{13}$ 10.

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The volume of a cubical box is 474.552 cubic metres. Find the length of each side of the box.

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Three numbers are to one another $2: 3: 4$. The sum of their cubes is 0.334125 . Find the numbers.

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Find the side of a cube whose volume is $\frac{24389}{216} \mathrm{~m}^{3}$.

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Evaluate:

(i) $\sqrt[3]{36} \times \sqrt[3]{384}$

(ii) $\sqrt[3]{96} \times \sqrt[3]{144}$

(iii) $\sqrt[3]{100} \times \sqrt[3]{270}$

(iv) $\sqrt[3]{121} \times \sqrt[3]{297}$

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Find the cube roots of the numbers $3048625,20346417,210644875,57066625$ using the fact that

(i) $3048625=3375 \times 729$

(ii) $20346417=9261 \times 2197$

(iii) $210644875=42875 \times 4913$

(iv) $57066625=166375 \times 343$

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Find the units digit of the cube root of the following numbers:

(i) 226981

(ii) 13824

(iii) 571787

(iv) 175616

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Find the tens digit of the cube root of each of the numbers in Q. No. 15 .

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## Exercise 4.5 - Cubes and Cube Roots | R.D. Sharma | Mathematics | Class 8

Making use of the cube root table, find the cube roots of the following (correct to three decimal places):(1-22)

7

70

700

7000

1100

780

7800

1346

250

5112

9800

732

42

133100

37800

0.27

8.6

0.86

8.65

7532

833

34.2

1) $1.913$

2) $4.121$

3) To find the cube root of $700$, let's first break $700$ into prime factors.

$$ 700 = 2^2×5^2×7 $$

$$ \sqrt[3]{700} = \sqrt[3]{2^2 \times 5^2 \times 7} $$

$$ \sqrt[3]{700} = \sqrt[3]{4} \times \sqrt[3]{25} \times \sqrt[3]{7} $$

$$ \sqrt[3]{700} = 1.578 \times 2.924 \times 1.913 $$

$$ \sqrt[3]{700} = 8.826 $$

4) To find the cube root of $7000$, let's first break $7000$ into simpler terms using prime factorization.

$$ 7000 = 2^3 \times 5^3 \times 7 $$

Now, extracting the cube root:

$$ \sqrt[3]{7000} = \sqrt[3]{2^3 \times 5^3 \times 7} $$

$$ \sqrt[3]{7000} = \sqrt[3]{2^3} \times \sqrt[3]{5^3} \times \sqrt[3]{7} $$

$$ \sqrt[3]{7000} = 2 \times 5 \times \sqrt[3]{7} $$

To find the cube root of $7$, referring to the cube root table:

$$ \sqrt[3]{7} \approx 1.913 $$

Thus,

$$ \sqrt[3]{7000} \approx 2 \times 5 \times 1.913 = 19.13 $$

So, the approximate cube root of 7000 is $19.13$.

5) To find the cube root of $1100$, let's first break it into simpler terms using prime factorization.

$$ 1100 = 2^2 \times 5^2 \times 11 $$

Now, extracting the cube root:

$$ \sqrt[3]{1100} = \sqrt[3]{2^2 \times 5^2 \times 11} $$

$$ \sqrt[3]{1100} = \sqrt[3]{4} \times \sqrt[3]{25} \times \sqrt[3]{11} $$

$$ \sqrt[3]{1100} = 1.587 \times 2.924 \times 2.224 $$

Thus,

$$ \sqrt[3]{1100} = 1.587 \times 2.924 \times 2.224 \approx 10.320 $$

So, the approximate cube root of 1100 is $10.320$.

6) To find the cube root of $780$, we first break $780$ into its prime factors:

$$ 780 = 2^2 \times 3 \times 5 \times 13 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{780} = \sqrt[3]{2^2 \times 3 \times 5 \times 13} $$

$$ \sqrt[3]{780} = \sqrt[3]{4} \times \sqrt[3]{3} \times \sqrt[3]{5} \times \sqrt[3]{13} $$

Computing these individual cube roots using a cube root table or estimation:

$$ \sqrt[3]{780} \approx \sqrt[3]{4} \times \sqrt[3]{3} \times \sqrt[3]{5} \times \sqrt[3]{13} $$

$$ \approx 1.587 \times 1.442 \times 1.710 \times 2.351 $$

Multiplying these values:

$$ \sqrt[3]{780} \approx 5.380 $$

7) To find the cube root of $7800$, we first break $7800$ into its prime factors:

$$ 7800 = 2^3 \times 3 \times 5^2 \times 13 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{7800} = \sqrt[3]{2^3 \times 3 \times 5^2 \times 13} $$

$$ \sqrt[3]{7800} = \sqrt[3]{8} \times \sqrt[3]{3} \times \sqrt[3]{25} \times \sqrt[3]{13} $$

Computing these individual cube roots using a cube root table or estimation:

$$ \sqrt[3]{7800} = 2 \times \sqrt[3]{3} \times \sqrt[3]{25} \times \sqrt[3]{13} $$

$$ = 2 \times 1.442 \times 2.924 \times 2.351 $$

Multiplying these values:

$$ \sqrt[3]{7800} \approx 9.912 $$

8) To find the cube root of (1346), let's first break it down into its prime factors:

$$ 1346 = 2 \times 673 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{1346} = \sqrt[3]{2} \times \sqrt[3]{673} $$

As 673 is a large prime number not readily available in standard cube root tables, we can provide a rough estimation:

$$ \sqrt[3]{1346} \approx \sqrt[3]{2} \times \sqrt[3]{673} $$

$$ \approx 1.260 \times \sqrt[3]{673} $$

We will estimate the value of $\sqrt[3]{673}$ to finalize the calculation.

The cube root of $673$ is approximately:

$$ \sqrt[3]{673} \approx 8.763 $$

Combining this with the cube root of $2$:

$$ \sqrt[3]{1346} \approx 1.260 \times 8.763 $$

Multiplying these values:

$$ \sqrt[3]{1346} \approx 11.041 $$

This is the approximate value of the cube root of 1346 using basic cube root table estimations and further precise calculation for the large prime component.

9) To find the cube root of (250), let's break it down into its prime factors:

$$ 250 = 2 \times 5^3 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{250} = \sqrt[3]{2 \times 5^3} $$

$$ \sqrt[3]{250} = \sqrt[3]{2} \times \sqrt[3]{125} $$

Now, since $\sqrt[3]{125} = 5$ (as (125) is a perfect cube), and using an estimation for $\sqrt[3]{2}$:

$$ \sqrt[3]{250} = 1.260 \times 5 = 6.3 $$

This is the approximate cube root of 250 using basic cube root table approximations.

10) To find the cube root of (5112), let's first break it down into its prime factors:

$$ 5112 = 2^4 \times 11 \times 29 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{5112} = \sqrt[3]{2^4 \times 11 \times 29} $$

$$ \sqrt[3]{5112} = \sqrt[3]{16} \times \sqrt[3]{11} \times \sqrt[3]{29} $$

Now taking the approximate cube roots using a table:

$$ \sqrt[3]{5112} \approx \sqrt[3]{16} \times \sqrt[3]{11} \times \sqrt[3]{29} $$

$$ \approx 2.5 \times 2.224 \times 3.072 $$

Multiplying these values:

$$ \sqrt[3]{5112} \approx 17.080 $$

11) To find the cube root of (9800), we first break it into its prime factors:

$$ 9800 = 2^3 \times 5^2 \times 7^2 $$

We then apply the cube root to each factor:

$$ \sqrt[3]{9800} = \sqrt[3]{2^3 \times 5^2 \times 7^2} $$

$$ \sqrt[3]{9800} = \sqrt[3]{8} \times \sqrt[3]{25} \times \sqrt[3]{49} $$

Calculating the individual cube roots using a cube root table or estimation:

$$ \sqrt[3]{9800} = 2 \times \sqrt[3]{25} \times \sqrt[3]{49} $$

$$ \approx 2 \times 2.924 \times 3.659 $$

Multiplying these values:

$$ \sqrt[3]{9800} = 2 \times 2.924 \times 3.659 \approx 21.397 $$

12) To find the cube root of (732), let's first factorize it:

$$ 732 = 2^2 \times 3 \times 61 $$

Now, we'll find the cube root of each factor:

$$ \sqrt[3]{732} = \sqrt[3]{2^2 \times 3 \times 61} $$

$$ = \sqrt[3]{4} \times \sqrt[3]{3} \times \sqrt[3]{61} $$

Using a cube root table or estimation for these individual components:

$$ \sqrt[3]{4} \approx 1.587 $$

$$ \sqrt[3]{3} \approx 1.442 $$

$$ \sqrt[3]{61} \approx 3.396 $$

Multiplying these values gives:

$$ \sqrt[3]{732} \approx 1.587 \times 1.442 \times 3.936 = 9.007 $$

Thus, the approximate cube root of 732 using the cube root table is about 9.007.

13) To calculate the cube root of (42), we start by expressing it in terms of its prime factors:

$$ 42 = 2 \times 3 \times 7 $$

Applying the cube root to each factor:

$$ \sqrt[3]{42} = \sqrt[3]{2} \times \sqrt[3]{3} \times \sqrt[3]{7} $$

Using cube root values from a cube root table or estimation:

$$ \sqrt[3]{2} \approx 1.260 $$

$$ \sqrt[3]{3} \approx 1.442 $$

$$ \sqrt[3]{7} \approx 1.913 $$

Now, let's compute the multiplication:

$$ \sqrt[3]{42} \approx 1.260 \times 1.442 \times 1.913 = 3.475 $$

Therefore, the approximate cube root of 42 using the cube root table is around (3.475).

14) To find the cube root of (133100), begin by factoring it:

$$ 133100 = 2^2 \times 5^2 \times 7^2 \times 43 $$

Applying the cube root operation:

$$ \sqrt[3]{133100} = \sqrt[3]{4} \times \sqrt[3]{25} \times \sqrt[3]{49} \times \sqrt[3]{43} $$

Using a cube root table or estimation:

$$ \sqrt[3]{4} \approx 1.587 $$

$$ \sqrt[3]{25} \approx 2.924 $$

$$ \sqrt[3]{49} \approx 3.659 $$

$$ \sqrt[3]{43} \approx 3.503 $$

Now, multiplying all these values:

$$ \sqrt[3]{133100} \approx 1.587 \times 2.924 \times 3.659 \times 3.504 = 59.495 $$

Thus, the approximate cube root of 133100 using a cube root table is about (59.495).

15) To calculate the cube root of (37800), we should start by factoring it:

$$ 37800 = 2^3 \times 3^3 \times 5^2 \times 7 $$

Applying the cube root to each factor:

$$ \sqrt[3]{37800} = \sqrt[3]{8} \times \sqrt[3]{27} \times \sqrt[3]{25} \times \sqrt[3]{7} $$

Using known cube roots from a cube root table or approximation:

$ \sqrt[3]{8} = 2 $ (since (8) is a perfect cube)

$ \sqrt[3]{27} = 3 $ (since (27) is a perfect cube)

$ \sqrt[3]{25} \approx 2.924 $ (using approximation)

$ \sqrt[3]{7} \approx 1.913 $ (using approximation)

Now, let's compute the multiplication:

$$ \sqrt[3]{37800} \approx 2 \times 3 \times 2.924 \times 1.913 = 33.561 $$

Therefore, the approximate cube root of (37800) using a cube root table is around (33.561).

16) To find the cube root of (0.27), let's express it in a form that's easier to manage with typical cube root tables:

$$ 0.27 = \frac{27}{100} $$

Now, factorize (27) and (100) into their prime factors:

$27 = 3^3$

$100 = 2^2 \times 5^2$

Applying the cube root:

$$ \sqrt[3]{0.27} = \sqrt[3]{\frac{27}{100}} = \frac{\sqrt[3]{27}}{\sqrt[3]{100}} $$

Using known cube roots:

$ \sqrt[3]{27} = 3 $ (since $27 = 3^3$)

$ \sqrt[3]{100} = \sqrt[3]{4 \times 25} = \sqrt[3]{4} \times \sqrt[3]{25} \approx 1.587 \times 2.924 $

Let's compute those:

$$ \sqrt[3]{100} \approx 1.587 \times 2.924 \approx 4.64 $$

Finally, calculating the cube root of (0.27):

$$ \sqrt[3]{0.27} \approx \frac{3}{4.64} \approx 0.647 $$

Therefore, the approximate cube root of (0.27) using a cube root table is around (0.647).

17) To find the cube root of (8.6), we can approximate directly as it’s a single small number not easily factorized into smaller components or perfect cubes.

Using estimation:

$$ \sqrt[3]{8.6} \approx 2.05 $$

This is based on the fact that:

( \sqrt[3]{8} = 2 ) (since (8 = 2^3))

( \sqrt[3]{9} = 2.08 ) (since (9) is close to a perfect cube)

Given that (8.6) is between (8) and (9), the cube root of (8.6) would be slightly above (2), but less than (2.08). Thus, approximating it as (2.05) makes sense based on these nearby cube root values.

18) To compute the cube root of (0.86) expressed as a fraction, we can write it as:

$$ 0.86 = \frac{86}{100} $$

Starting with factorizing the numerator and the denominator:

$ 86 = 2 \times 43 $

$ 100 = 2^2 \times 5^2 $

Applying the cube root:

$$ \sqrt[3]{0.86} = \sqrt[3]{\frac{86}{100}} = \frac{\sqrt[3]{86}}{\sqrt[3]{100}} $$

Now using cube root values or estimations:

$ \sqrt[3]{86} $ is approximately $4.414$

$ \sqrt[3]{100} = \sqrt[3]{4 \times 25} = \sqrt[3]{4} \times \sqrt[3]{25} \approx 1.587 \times 2.924 $ which we computed earlier as approximately (4.64).

Then, calculating the cube root of (0.86):

$$ \sqrt[3]{0.86} = \frac{4.414}{4.64} \approx 0.951 $$

19) To compute the cube root of (8.65), written as a fraction, we can express it as:

$$ 8.65 = \frac{865}{100} $$

Now we factorize:

( 865 = 5 \times 173 )

( 100 = 2^2 \times 5^2 )

Applying the cube root to each term:

$$ \sqrt[3]{8.65} = \sqrt[3]{\frac{865}{100}} = \frac{\sqrt[3]{865}}{\sqrt[3]{100}} $$

The cube roots of the factors are roughly:

$ \sqrt[3]{865} $ (as (865) is not close to any perfect cubes, an estimation or cube root table reference is needed)

$ \sqrt[3]{100} \approx 4.64 $ as previously calculated

To estimate $ \sqrt[3]{865} $:

$ \sqrt[3]{865} \approx 9.54 $ (based on direct calculation or approximation)

Then, computing the cube root of (8.65):

$$ \sqrt[3]{8.65} \approx \frac{9.54}{4.64} \approx 2.056 $$

This value (2.056) is based on these approximations, representing the estimated cube root of (8.65) written as (865/100), using cube root table concepts for basic component factors.

20) To find the cube root of (7532), let's break it down into its prime factors:

$$ 7532 = 2^2 \times 1883 $$

Now we apply the cube root to each factor:

$$ \sqrt[3]{7532} = \sqrt[3]{2^2 \times 1883} $$ $$ = \sqrt[3]{4} \times \sqrt[3]{1883} $$

Using estimations from cube root tables:

$ \sqrt[3]{4} \approx 1.587 $

Calculating or estimating the cube root of (1883) (as it’s not a commonly tabulated number):

$ \sqrt[3]{1883} \approx 12.34 $ (based on precise computation or close placement with known cube values)

Now, multiplying these values: $$ \sqrt[3]{7532} \approx 1.587 \times 12.34 \approx 19.602 $$

Thus, the approximate cube root of (7532) using a cube root table is around (19.602).

21) To compute the cube root of (833), let’s first factorize it:

$$ 833 = 7 \times 7 \times 17 $$

Applying the cube root to each factor:

$$ \sqrt[3]{833} = \sqrt[3]{7^2 \times 17} $$ $$ = \sqrt[3]{49} \times \sqrt[3]{17} $$

Using cube root estimations from available values:

$ \sqrt[3]{49} \approx 3.659 $ (since (49) is (7^2) and the cube root of (7) can be approximated)

$ \sqrt[3]{17} \approx 2.571 $

Multiplying these values:

$$ \sqrt[3]{833} \approx 3.659 \times 2.571 \approx 9.407 $$

Therefore, the approximate cube root of (833) using cube root table values is around (9.407).

22) To calculate the cube root of (34.2), we can approach it directly since (34.2) isn't easily factorable into simpler components with known cube roots. We'll need to use an estimation based on cube root tables or direct calculation:

Given $ \sqrt[3]{27} = 3 $ (because $27 = 3^3$), and $ \sqrt[3]{64} = 4 $ (because $64 = 4^3$), (34.2) resides between the cubes of these numbers.

Estimating $ \sqrt[3]{34.2} $ by considering the proximity to these cubes:

(34.2) is closer to (27) than (64), which suggests a cube root slightly higher than (3).

An approximate cube root using estimation: $$ \sqrt[3]{34.2} \approx 3.27 $$

What is the length of the side of a cube whose volume is $275 \mathrm{~cm}^{3}$. Make use of the table for the cube root, and explain with steps.