Squares and Square Roots - Class 8 - Mathematics
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Examples - Squares and Square Roots | NCERT | Mathematics | Class 8
Find the square of the following numbers without actual multiplication.
(i) 39
(ii) 42
To find the square of numbers without actual multiplication, we can use the identity for the square of a number near a base that is easy to square (such as 10, 20, 30, etc.). Let's use the following identities based on how close the given numbers are to a convenient base:
For 39 (near 40)
We can use the identity $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 40$ and $b = 1$ since $39 = 40 - 1$.
For 42 (near 40)
Similarly, we can use the identity $(a + b)^2 = a^2 + 2ab + b^2$, where $a = 40$ and $b = 2$ since $42 = 40 + 2$.
Let's perform these calculations.
(i) For 39
Using the formula ( (a - b)^2 ): $$ $40 - 1$^2 = 40^2 - 2 \cdot 40 \cdot 1 + 1^2 $$
(ii) For 42
Using the formula ( (a + b)^2 ): $$ $40 + 2$^2 = 40^2 + 2 \cdot 40 \cdot 2 + 2^2 $$
Let's compute these specific values.
For $40^2$(common in both cases),
Subtraction and addition operations for both cases.
Calculation
$40^2$ is $1600$.
(i) For 39:
$$ 1600 - 80 + 1 = 1521 $$
(ii) For 42:
$$ 1600 + 160 + 4 = 1764 $$
So, the squares of the numbers are:
$39^2 = 1521$
$42^2 = 1764$
Write a Pythagorean triplet whose smallest member is 8.
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Sign up nowFind a Pythagorean triplet in which one member is 12.
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Sign up nowFind the square root of 6400.
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Sign up nowIs 90 a perfect square?
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Sign up nowIs 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.
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Sign up nowFind the smallest number by which 9408 must be divided so that the quotient is a perfect square. Find the square root of the quotient.
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Sign up nowFind the smallest square number which is divisible by each of the numbers 6,9 and 15
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Sign up nowFind the square root of :
(i) 729
(ii) 1296
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Sign up nowFind the least number that must be subtracted from 5607 so as to get a perfect square. Also find the square root of the perfect square.
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Sign up nowFind the greatest 4-digit number which is a perfect square.
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Sign up nowFind the least number that must be added to 1300 so as to get a perfect square. Also find the square root of the perfect square.
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Sign up nowFind the square root of 12.25.
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Sign up nowArea of a square plot is $2304 \mathrm{~m}^{2}$. Find the side of the square plot.
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Sign up nowThere are 2401 students in a school. P.T. teacher wants them to stand in rows and columns such that the number of rows is equal to the number of columns. Find the number of rows.
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Sign up nowExercise 5.1 - Squares and Square Roots | NCERT | Mathematics | Class 8
What will be the unit digit of the squares of the following numbers?
(i) 81
(ii) 272
(iii) 799
(iv) 3853
(v) 1234
(vi) 26387
(vii) 52698
(viii) 99880
(ix) 12796
(x) 55555
To determine the unit digit of the squares of the given numbers, we can square the unit digit of each number and then look at the unit digit of the result. The pattern of the unit digits of squares is cyclical because of the properties of multiplication in base-10 arithmetic. Here's the pattern for each digit when squared:
$0^2 = 0$
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 6$
$5^2 = 5$
$6^2 = 6$
$7^2 = 9$
$8^2 = 4$
$9^2 = 1$
Using this pattern, we can find the unit digit of the square of each given number:
For 81, the unit digit is 1. $1^2 = 1$.
For 272, the unit digit is 2. $2^2 = 4$.
For 799, the unit digit is 9. $9^2 = 1$.
For 3853, the unit digit is 3. $3^2 = 9$.
For 1234, the unit digit is 4. $4^2 = 6$.
For 26387, the unit digit is 7. $7^2 = 9$.
For 52698, the unit digit is 8. $8^2 = 4$.
For 99880, the unit digit is 0. $0^2 = 0$.
For 12796, the unit digit is 6. $6^2 = 6$.
For 55555, the unit digit is 5. $5^2 = 5$.
So, the unit digits of the squares of the given numbers are:
(i) 1, (ii) 4, (iii) 1, (iv) 9, (v) 6, (vi) 9, (vii) 4, (viii) 0, (ix) 6, (x) 5.
The following numbers are obviously not perfect squares. Give reason.
(i) 1057
(ii) 23453
(iii) 7928
(iv) 222222
(v) 64000
(vi) 89722
(vii) 222000
(viii) 505050
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Sign up nowThe squares of which of the following would be odd numbers?
(i) 431
(ii) 2826
(iii) 7779
(iv) 82004
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Sign up nowObserve the following pattern and find the missing digits.
$$
\begin{aligned}
11^{2} & =121 \\
101^{2} & =10201 \\
1001^{2} & =1002001 \\
100001^{2} & =1 \ldots \ldots . .2 . \\
10000001^{2} & =\ldots \ldots \ldots \ldots . . . . .
\end{aligned}
$$
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Sign up nowObserve the following pattern and supply the missing numbers.
$$
\begin{aligned}
& 11^{2}=121 \\
& 101^{2}=10201 \\
& 10101^{2}=102030201 \\
& 1010101^{2}= \\
& .^{2}=10203040504030201
\end{aligned}
$$
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Sign up nowUsing the given pattern, find the missing numbers.
$1^{2}+2^{2}+2^{2}=3^{2}$
$2^{2}+3^{2}+6^{2}=7^{2}$
$3^{2}+4^{2}+12^{2}=13^{2}$
$4^{2}+5^{2}+{ }^{2}=21^{2}$
$5^{2}+{ }^{2}+30^{2}=31^{2}$
$6^{2}+7^{2}++_{-}^{2}=-^{2}$
## To find pattern
Third number is related to first and second number. How?
Fourth number is related to third number. How?
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Sign up nowWithout adding, find the sum.
(i) $1+3+5+7+9$
(ii) $1+3+5+7+9+11+13+15+17+19$
(iii) $1+3+5+7+9+11+13+15+17+19+21+23$
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Sign up now(i) Express 49 as the sum of 7 odd numbers.
(ii) Express 121 as the sum of 11 odd numbers.
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Sign up nowHow many numbers lie between squares of the following numbers?
(i) 12 and 13
(ii) 25 and 26
(iii) 99 and 100
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Sign up nowExercise 5.2 - Squares and Square Roots | NCERT | Mathematics | Class 8
Find the square of the following numbers.
(i) 32
(ii) 35
(iii) 86
(iv) 93
(v) 71
(vi) 46
The squares of the given numbers are:
(i) $32^2 = 1024$
(ii) $35^2 = 1225$
(iii) $86^2 = 7396$
(iv) $93^2 = 8649$
(v) $71^2 = 5041$
(vi) $46^2 = 2116$
Write a Pythagorean triplet whose one member is.
(i) 6 (ii) 14 (iii) 16 (iv) 18
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Sign up nowExercise 5.3 - Squares and Square Roots | NCERT | Mathematics | Class 8
What could be the possible 'one's' digits of the square root of each of the following numbers?
(i) 9801
(ii) 99856
(iii) 998001
(iv) 657666025
The 'one's' digits of the square root of each of the given numbers are as follows:
- 9801: The square root is 99, so the 'one's' digit is 9.
- 99856: The square root is 316, so the 'one's' digit is 6.
- 998001: The square root is 999, so the 'one's' digit is 9.
- 657666025: The square root is 25645, so the 'one's' digit is 5.
Without doing any calculation, find the numbers which are surely not perfect squares.
(i) 153
(ii) 257
(iii) 408
(iv) 441
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Sign up nowFind the square roots of 100 and 169 by the method of repeated subtraction.
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Sign up nowFind the square roots of the following numbers by the Prime Factorisation Method.
(i) 729
(ii) 400
(iii) 1764
(iv) 4096
(v) 7744
(vi) 9604
(vii) 5929
(viii) 9216
(ix) 529
(x) 8100
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Sign up nowFor each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
(i) 252
(ii) 180
(iii) 1008
(iv) 2028
(v) 1458
(vi) 768
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Sign up nowFor each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.
(i) 252
(ii) 2925
(iii) 396
(iv) 2645
(v) 2800
(vi) 1620
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Sign up nowThe students of Class VIII of a school donated ₹ 2401 in all, for Prime Minister's National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.
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Sign up now2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
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Sign up nowFind the smallest square number that is divisible by each of the numbers 4,9 and 10.
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Sign up nowFind the smallest square number that is divisible by each of the numbers 8,15 and 20.
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Sign up nowExercise 5.4 - Squares and Square Roots | NCERT | Mathematics | Class 8
Find the square root of each of the following numbers by Division method.
(i) 2304
(ii) 4489
(iii) 3481
(iv) 529
(v) 3249
(vi) 1369
(vii) 5776
(viii) 7921
(ix) 576
(x) 1024
(xi) 3136
(xii) 900
Find the number of digits in the square root of each of the following numbers (without any calculation).
(i) 64
(ii) 144
(iii) 4489
(iv) 27225
(v) 390625
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Sign up nowFind the square root of the following decimal numbers.
(i) 2.56
(ii) 7.29
(iii) 51.84
(iv) 42.25
(v) 31.36
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Sign up nowFind the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i) 402
(ii) 1989
(iii) 3250
(iv) 825
(v) 4000
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Sign up nowFind the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.
(i) 525
(ii) 1750
(iii) 252
(iv) 1825
(v) 6412
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Sign up nowFind the length of the side of a square whose area is $441 \mathrm{~m}^{2}$.
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Sign up nowIn a right triangle $\mathrm{ABC}, \angle \mathrm{B}=90^{\circ}$.
(a) If $\mathrm{AB}=6 \mathrm{~cm}, \mathrm{BC}=8 \mathrm{~cm}$, find $\mathrm{AC}$
(b) If $\mathrm{AC}=13 \mathrm{~cm}, \mathrm{BC}=5 \mathrm{~cm}$, find $\mathrm{AB}$
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Sign up nowA gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain same. Find the minimum number of plants he needs more for this.
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Sign up nowThere are 500 children in a school. For a P.T. drill they have to stand in such a manner that the number of rows is equal to number of columns. How many children would be left out in this arrangement.
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Sign up nowExtra Questions - Squares and Square Roots | NCERT | Mathematics | Class 8
Find the sum of the squares of the following: $\frac{\sqrt{3}}{\sqrt{2}+1}$, $\frac{\sqrt{3}}{\sqrt{2}-1}$, $\frac{\sqrt{2}}{\sqrt{3}}$.
A) $\frac{56}{4}$
B) $18 \frac{2}{3}$
C) $2 \frac{18}{3}$
D) $3 \frac{18}{2}$
To determine the sum of the squares of the given expressions $\frac{\sqrt{3}}{\sqrt{2}+1}$, $\frac{\sqrt{3}}{\sqrt{2}-1}$, and $\frac{\sqrt{2}}{\sqrt{3}}$, we proceed as follows:
Calculating $\left(\frac{\sqrt{3}}{\sqrt{2}+1}\right)^{2}$: $$ \left(\frac{\sqrt{3}}{\sqrt{2}+1}\right)^{2} = \frac{3}{2 + 1 + 2\sqrt{2}} = \frac{3}{3 + 2\sqrt{2}} $$
Calculating $\left(\frac{\sqrt{3}}{\sqrt{2}-1}\right)^{2}$: $$ \left(\frac{\sqrt{3}}{\sqrt{2}-1}\right)^{2} = \frac{3}{2 + 1 - 2\sqrt{2}} = \frac{3}{3 - 2\sqrt{2}} $$
Calculating $\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2}$: $$ \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2} = \frac{2}{3} $$
Adding them together: $$ \begin{align*} \left(\frac{\sqrt{3}}{\sqrt{2}+1}\right)^{2} + \left(\frac{\sqrt{3}}{\sqrt{2}-1}\right)^{2} + \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2} &= \frac{3}{3 + 2\sqrt{2}} + \frac{3}{3 - 2\sqrt{2}} + \frac{2}{3} \ &= \frac{3(3 - 2\sqrt{2}) + 3(3 + 2\sqrt{2})}{9 - 4} + \frac{2}{3} \ &= \frac{9 - 6\sqrt{2} + 9 + 6\sqrt{2}}{5} + \frac{2}{3} \ &= \frac{18}{1} + \frac{2}{3} = 18 + \frac{2}{3} = 18 \frac{2}{3} \end{align*} $$
The correct option is B $18 \frac{2}{3}$.
"The smallest number by which 252 must be multiplied so that the resultant is a perfect square is:"
A) 2
B) 3
C) 5
D) 7
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Sign up nowSum of 6 consecutive integers is 3. Find the sum of squares of those 6 integers.
A. 19
B. 12
C. 18
D. 20
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Sign up nowThe difference between the squares of two numbers is 45. The square of the smaller number is 4 times the larger number. Find the numbers.
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Sign up nowProve that if $x$ and $y$ are odd positive integers, then $x^2 + y^2$ is even but not divisible by 4.
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Sign up nowA square $ABCD$ of diagonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at right angles. The shortest distance between $DC$ and $AB$ is:
A) $\sqrt{2}a$
B) $\frac{2a}{\sqrt{3}}$
C) $\frac{2a}{\sqrt{5}}$
D) $\left(\frac{\sqrt{3}}{2}\right)a$
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