Powers - Class 8 - Mathematics
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Examples - Powers | R.D. Sharma | Mathematics | Class 8
Express each of the following as a rational number of the form $\frac{p}{q}$ :
(i) $5^{-3}$
(ii) $(-2)^{-5}$
(iii) $\left(\frac{4}{3}\right)^{-3}$
(iv) $\left(\frac{-2}{5}\right)^{-4}$
(v) $\frac{1}{2^{-3}}$
To express each of the given expressions as a rational number of the form $\frac{p}{q}$, we use the property that $a^{-n} = \frac{1}{a^n}$ for any nonzero $a$ and positive integer $n$. Therefore:
(i) $$5^{-3} = \frac{1}{5^3} = \frac{1}{125}$$
(ii) $$(-2)^{-5} = \frac{1}{(-2)^5} = \frac{1}{-32}$$
(iii) $$\left(\frac{4}{3}\right)^{-3} = \frac{1}{\left(\frac{4}{3}\right)^3} = \frac{1}{\frac{64}{27}} = \frac{27}{64}$$
(iv) $$\left(\frac{-2}{5}\right)^{-4} = \frac{1}{\left(\frac{-2}{5}\right)^4} = \frac{1}{\frac{16}{625}} = \frac{625}{16}$$
(v) $$\frac{1}{2^{-3}} = 2^3 = 8$$
Hence, the rational expressions are:
(i) $\frac{1}{125}$
(ii) $\frac{1}{-32}$
(iii) $\frac{27}{64}$
(iv) $\frac{625}{16}$
(v) $8$, which can also be written as $\frac{8}{1}$ for consistency with the $\frac{p}{q}$ form.
Express each of the following as a rational number of the form $\frac{p}{q}$ :
(i) $\left(\frac{3}{8}\right)^{-2} \times\left(\frac{4}{5}\right)^{-3}$
(ii) $\left(\frac{-2}{7}\right)^{-4} \times\left(\frac{-7}{5}\right)^{2}$
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Sign up nowExpress each of the following as power of a rational number with positiv exponent:
(i) $\left(\frac{1}{4}\right)^{-3}$
(ii) $5^{-3} \times 5^{-6}$
(iii) $\left(\frac{-1}{4}\right)^{-5} \times\left(\frac{-1}{4}\right)^{-7}$
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Sign up nowSimplify, with steps
(i) $\left(2^{-1}+5^{-1}\right)^{2} \times\left(\frac{-5}{8}\right)^{-1}$
(ii) $\left(6^{-1}-8^{-1}\right)^{-1}+\left(2^{-1}-3^{-1}\right)^{-1}$
(iii) $\left(5^{-1} \times 3^{-1}\right)^{-1}+6^{-1}$
(iv) $\left(4^{-1}+8^{-1}\right)+\left(\frac{2}{3}\right)^{-1}$
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Sign up nowSimplify, with steps
(i) $(\frac{1}{4})^{-2} + (\frac{1}{2})^{-2} + (\frac{1}{3})^{-2}$
(ii) $\left\{6^{-1} + (\frac{3}{2})^{-1}\right\}^{-1}$
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Sign up nowExpress each of the following as a rational number of the form $\frac{p}{q}$ :
(i) $\left(2^{-1}+3^{-1}\right)^{2}$
(ii) $\left(2^{-1}-4^{-1}\right)^{2}$
(iii) $\left\{\left(\frac{3}{4}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}$
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Sign up nowBy what number should $(-8)^{-1}$ be multiplied so that the product may be equal to $10^{-1}$ ?
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Sign up nowBy what number should $(-24)^{-1}$ be divided so that the quotient may be $3^{-1}$ ?
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Sign up nowUsing the laws of exponents, simplify each of the following and express in exponential form:
(i) $3^{7} \times 3^{-2}$
(ii) $2^{-7} \div 2^{-3}$
(iii) $\left(5^{2}\right)^{-3}$
(iv) $2^{-3} \times(-7)^{-3}$
(v) $\frac{3^{-5}}{4^{-5}}$
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Sign up nowUsing the laws of exponents simplify and express each of the following in exponential form with positive exponent, with steps
(i) $(-4)^{4} \times(-4)^{-10}$
(ii) $2^{-5}+2^{2}$
(iii) $3^{-4} \times 2^{-4}$
(iv) $\left(\frac{1}{2^{3}}\right)^{2}$
(v) $\left(3^{-7}+3^{-10}\right) \times 3^{-5}$
(vi) $(-3)^{4} \times\left(\frac{5}{3}\right)^{4}$
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Sign up nowSimplify and write the answer in the exponential form, with steps
(i) $\left(2^{5} \div 2^{8}\right)^{5} \times 2^{-5}$
(ii) $(-4)^{3} \times(5)^{-3} \times(-5)^{-3}$
(iii) $\frac{1}{8} \times 3^{-3}$
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Sign up nowSimplify each of the following:
(i) $\left[\left\{\left(\frac{-1}{5}\right)^{-2}\right\}^{2}\right]^{-1}$
(ii) $\left\{\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-2}$
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Sign up nowSimplify:
(i) $\left(\frac{5}{8}\right)^{-7} \times\left(\frac{8}{5}\right)^{-5}$
(ii) $\left(\frac{-2}{3}\right)^{-2} \times\left(\frac{4}{5}\right)^{-3}$
(iii) $\left(\frac{3}{4}\right)^{-4} \div\left(\frac{3}{2}\right)^{-3}$
(iv) $\left(\frac{3}{7}\right)^{-2} \times\left(\frac{7}{6}\right)^{-3}$
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Sign up nowEvaluate: $\frac{8^{-1} \times 5^{3}}{2^{-4}}$, with steps
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Sign up nowSimplify, with steps
(i) $\frac{25 \times a^{-4}}{5^{-3} \times 10 \times a^{-8}}$
(ii) $\frac{3^{-5} \times 10^{-5} \times 125}{5^{-7} \times 6^{-5}}$
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Sign up nowBy what number should $(-4)^{-2}$ be multiplied so that the product may be equal to $10^{-2}$?
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Sign up nowBy what number should $(-12)^{-1}$ be divided so that the quotient may be $\left(\frac{2}{3}\right)^{-1}$ ?
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Sign up nowBy what number should $\left(\frac{-3}{2}\right)^{-3}$ be divided so that the quotient may be
$$ \left(\frac{4}{27}\right)^{-2} ? $$
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Sign up nowFind $x$ so that $\left(\frac{5}{3}\right)^{-5} \times\left(\frac{5}{3}\right)^{-11}=\left(\frac{5}{3}\right)^{8 x}$
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Sign up nowFind $m$ so that $\left(\frac{2}{9}\right)^{3} \times\left(\frac{2}{9}\right)^{-6}=\left(\frac{2}{9}\right)^{2 m-1}$
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Sign up nowIf $x=\left(\frac{3}{2}\right)^{2} \times\left(\frac{2}{3}\right)^{-4}$, find the value of $x^{-2}$, with steps
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Sign up nowWrite the following numbers in standard form:
(i) 0.4579
(ii) 0.000007
(iii) 0.000000564
(iv) 0.0000021
(v) 216000000
(vi) $0.0000529 \times 10^{4}$
(vii) $9573 \times 10^{-4}$
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Sign up nowExpress the following numbers in usual form:
(i) $3.52 \times 10^{5}$
(ii) $7.54 \times 10^{-4}$
(iii) $3 \times 10^{-5}$
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Sign up nowExpress the number appearing in the following statements in standard form:
(i) 1 micron is equal to $\frac{1}{1000000}$ metre.
(ii) Charge of an electron is 0.0000000000000000016 coloumbs.
(iii) Size of a bacteria is 0.0000005 metre.
(iv) Size of a plant cell is 0.00001275 metre.
(v) Thickness of a normal paper is $0.07 \mathrm{~mm}$.
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Sign up nowIf the diameters of the Sun and the Earth are $1.4 \times 10^{9}$ metres and $1.275 \times 10^{7}$ metres respectively. Compare these two.
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Sign up nowThe size of a red blood cell is $0.000007 \mathrm{~m}$ and the size of a plant cell is $0.00001275 \mathrm{~m}$. Compare these two.
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Sign up nowExercise 2.1 - Powers | R.D. Sharma | Mathematics | Class 8
Express each of the following as a rational number of the form $\frac{p}{q}$, where $p$ and $q$ ar integers and $q \neq 0$ :
(i) $2^{-3}$
(ii) $(-4)^{-2}$
(iii) $\frac{1}{3^{-2}}$
(iv) $\left(\frac{1}{2}\right)^{-5}$
(v) $\left(\frac{2}{3}\right)^{-2}$
Here are the expressions expressed as rational numbers of the form $\frac{p}{q}$:
For $2^{-3}$, it is expressed as $\frac{1}{8}$.
For $(-4)^{-2}$, it is expressed as $\frac{1}{16}$.
For $\frac{1}{3^{-2}}$, it is expressed as $9$ or $\frac{9}{1}$.
For $\left(\frac{1}{2}\right)^{-5}$, it is expressed as $32$ or $\frac{32}{1}$.
For $\left(\frac{2}{3}\right)^{-2}$, it is expressed as $\frac{9}{4}$.
Find the values of each of the following, with steps:
(i) $3^{-1}+4^{-1}$
(ii) $\left(3^{0}+4^{-1}\right) \times 2^{2}$
(iii) $\left(3^{-1}+4^{-1}+5^{-1}\right)^{0}$
(iv) $\left\{\left(\frac{1}{3}\right)^{-1}-\left(\frac{1}{4}\right)^{-1}\right\}^{-1}$
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Sign up nowFind the values of each of the following, with steps:
(i) $\left(\frac{1}{2}\right)^{-1}+\left(\frac{1}{3}\right)^{-1}+\left(\frac{1}{4}\right)^{-1}$
(ii) $\left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2}$
(iii) $\left(2^{-1} \times 4^{-1}\right) \div 2^{-2}$
(iv) $\left(5^{-1} \times 2^{-1}\right) \div 6^{-1}$
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Sign up nowSimplify, with steps for each:
(i) $\left(4^{-1} \times 3^{-1}\right)^{2}$
(ii) $\left(5^{-1} \div 6^{-1}\right)^{3}$
(iii) $\left(2^{-1}+3^{-1}\right)^{-1}$
(iv) $\left(3^{-1} \times 4^{-1}\right)^{-1} \times 5^{-1}$
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Sign up nowSimplify, with steps for each:
(i) $\left(3^{2}+2^{2}\right) \times\left(\frac{1}{2}\right)^{3}$
(ii) $\left(3^{2}-2^{2}\right) \times\left(\frac{2}{3}\right)^{-3}$
(iii) $\left[\left(\frac{1}{3}\right)^{-3}-\left(\frac{1}{2}\right)^{-3}\right] \div\left(\frac{1}{4}\right)^{-3}$
(iv) $\left(2^{2}+3^{2}-4^{2}\right) \div\left(\frac{3}{2}\right)^{2}$
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Sign up nowBy what number should $5^{-1}$ be multiplied so that the product may be equal to $(-7)^{-1}$ ?
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Sign up nowBy what number should $\left(\frac{1}{2}\right)^{-1}$ be multiplied so that the product may be equal t $\left(-\frac{4}{7}\right)^{-1} ?$
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Sign up nowBy what number should $(-15)^{-1}$ be divided so that the quotient may be equal to $(-5)^{-1}$.
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Sign up nowExercise 2.2 - Powers | R.D. Sharma | Mathematics | Class 8
Write each of the following in exponential form:
(i) $\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1}$
(ii) $\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2}$
To write the given expressions in exponential form, we apply the rule of exponents that states if we multiply powers with the same base, we add the exponents. Let's simplify each expression:
(i) $\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1}$
Since we're multiplying the expression $\left(\frac{3}{2}\right)^{-1}$ four times, we add the exponents:
The result is $\left(\frac{3}{2}\right)^{-4}$.
(ii) $\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2}$
Similarly, we're multiplying the expression $\left(\frac{2}{5}\right)^{-2}$ three times, so we add the exponents:
The result is $\left(\frac{2}{5}\right)^{-6}$.
Evaluate, with steps
(i) $5^{-2}$
(ii) $(-3)^{-2}$
(iii) $\left(\frac{1}{3}\right)^{-4}$
(iv) $\left(\frac{-1}{2}\right)^{-1}$
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Sign up nowExpress each of the following as a rational number in the form $\frac{p}{q}$ :
(i) $6^{-1}$
(ii) $(-7)^{-1}$
(iii) $\left(\frac{1}{4}\right)^{-1}$
(iv) $(-4)^{-1} \times\left(\frac{-3}{2}\right)^{-1}$
(v) $\left(\frac{3}{5}\right)^{-1} \times\left(\frac{5}{2}\right)^{-1}$
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Sign up nowSimplify, with steps
(i) $\left\{4^{-1} \times 3^{-1}\right\}^{2}$
(ii) $\left\{5^{-1} \div 6^{-1}\right\}^{3}$
(iii) $\left(2^{-1}+3^{-1}\right)^{-1}$
(iv) $\left\{3^{-1} \times 4^{-1}\right\}^{-1} \times 5^{-1}$
(v) $\left(4^{-1}-5^{-1}\right) \div 3^{-1}$
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Sign up nowExpress each of the following rational numbers with a negative exponent:
(i) $\left(\frac{1}{4}\right)^{3}$
(ii) $3^{5}$
(iii) $\left(\frac{3}{5}\right)^{4}$
(iv) $\left\{\left(\frac{3}{2}\right)^{4}\right\}^{-3}$
(v) $\left\{\left(\frac{7}{3}\right)^{4}\right\}^{-3}$
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Sign up nowExpress each of the following rational numbers with a positive exponent, with steps
(i) $\left(\frac{3}{4}\right)^{-2}$
(ii) $\left(\frac{5}{4}\right)^{-3}$
(iii) $4^{3} \times 4^{-9}$
(iv) $\left\{\left(\frac{4}{3}\right)^{-3}\right\}^{-4}$
(v) $\left\{\left(\frac{3}{2}\right)^{4}\right\}^{-2}$
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Sign up nowSimplify, with steps
(i) $\left\{\left(\frac{1}{3}\right)^{-3}-\left(\frac{1}{2}\right)^{-3}\right\} \div\left(\frac{1}{4}\right)^{-3}$
(ii) $\left(3^{2}-2^{2}\right) \times\left(\frac{2}{3}\right)^{-3}$
(iii) $\left\{\left(\frac{1}{2}\right)^{-1} \times(-4)^{-1}\right\}^{-1}$
(iv) $\left[\left\{\left(\frac{-1}{4}\right)^{2}\right\}^{-2}\right]^{-1}$
(v) $\left\{\left(\frac{2}{3}\right)^{2}\right\}^{3} \times\left(\frac{1}{3}\right)^{-4} \times 3^{-1} \times 6^{-1}$
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Sign up nowBy what number should $5^{-1}$ be multiplied so that the product may be equal to $(-7)^{-1}$ ?
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Sign up nowBy what number should $\left(\frac{1}{2}\right)^{-1}$ be multiplied so that the product may be equal to $\left(\frac{-4}{7}\right)^{-1} ?$
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Sign up nowBy what number should $(-15)^{-1}$ be divided so that the quotient may be equal to $(-5)^{-1}$ ?
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Sign up nowBy what number should $\left(\frac{5}{3}\right)^{-2}$ be multiplied so that the product may be $\left(\frac{7}{3}\right)^{-1}$ ?
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Sign up nowFind $x$, with steps, if
(i) $\left(\frac{1}{4}\right)^{-4} \times\left(\frac{1}{4}\right)^{-8}=\left(\frac{1}{4}\right)^{-4 x}$
(ii) $\left(\frac{-1}{2}\right)^{-19} \div\left(\frac{-1}{2}\right)^{8}=\left(\frac{-1}{2}\right)^{-2 x+1}$
(iii) $\left(\frac{3}{2}\right)^{-3} \times\left(\frac{3}{2}\right)^{5}=\left(\frac{3}{2}\right)^{2 x+1}$
(iv) $\left(\frac{2}{5}\right)^{-3} \times\left(\frac{2}{5}\right)^{15}=\left(\frac{2}{5}\right)^{2+3 x}$
(v) $\left(\frac{5}{4}\right)^{-x} \div\left(\frac{5}{4}\right)^{-4}=\left(\frac{5}{4}\right)^{5}$
(vi) $\left(\frac{8}{3}\right)^{2 x+1} \times\left(\frac{8}{3}\right)^{5}=\left(\frac{8}{3}\right)^{x+2}$
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Sign up now(i) If $x=\left(\frac{3}{2}\right)^{2} \times\left(\frac{2}{3}\right)^{-4}$, find the value of $x^{-2}$.
(ii) If $x=\left(\frac{4}{5}\right)^{-2} \div\left(\frac{1}{4}\right)^{2}$, find the value of $x^{-1}$.
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Sign up nowFind the value of $x$ for which $5^{2 x} \div 5^{-3}=5^{5}$.
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Sign up nowExercise 2.3 - Powers | R.D. Sharma | Mathematics | Class 8
Express the following numbers in standard form:
(i) 6020000000000000
(ii) 0.00000000000942
(iii) 0.00000000085
(iv) $846 \times 10^{7}$
(v) $3759 \times 10^{-4}$
(vi) 0.00072984
(vii) $0.000437 \times 10^{4}$
(viii) $4 \div 100000$
Here are the expressions in standard form:
(i) $6.02 \times 10^{15}$
(ii) $9.42 \times 10^{-12}$
(iii) $8.5 \times 10^{-10}$
(iv) $8.46 \times 10^{9}$
(v) $3.759 \times 10^{-1}$
(vi) $7.2984 \times 10^{-4}$
(vii) $4.37 \times 10^{0}$ (or simply 4.37)
(viii) $4.0 \times 10^{-5}$
Write the following numbers in the usual form:
(i) $4.83 \times 10^{7}$
(ii) $3.02 \times 10^{-6}$
(iii) $4.5 \times 10^{4}$
(iv) $3 \times 10^{-8}$
(v) $1.0001 \times 10^{9}$
(vi) $5.8 \times 10^{2}$
(vii) $3.61492 \times 10^{6}$
(viii) $3.25 \times 10^{-7}$
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Sign up nowExercise (MCQs) - Powers | R.D. Sharma | Mathematics | Class 8
Square of $\left(\frac{-2}{3}\right)$ is
(a) $-\frac{2}{3}$
(b) $\frac{2}{3}$
(c) $-\frac{4}{9}$
(d) $\frac{4}{9}$
To find the square of $\left(\frac{-2}{3}\right)$, we calculate $\left(\frac{-2}{3}\right)^2$.
$$ \left(\frac{-2}{3}\right)^2 = \frac{(-2)^2}{3^2} = \frac{4}{9} $$
Therefore, the correct answer is (d) $\frac{4}{9}$.
Cube of $\frac{-1}{2}$ is
(a) $\frac{1}{8}$
(b) $\frac{1}{16}$
(c) $-\frac{1}{8}$
(d) $\frac{-1}{16}$
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Sign up nowWhich of the following is not equal to $\left(\frac{-3}{5}\right)^{4}$ ?
(a) $\frac{(-3)^{4}}{5^{4}}$
(b) $\frac{3^{4}}{(-5)^{4}}$
(c) $-\frac{3^{4}}{5^{4}}$
(d) $\frac{-3}{5} \times \frac{-3}{5} \times \frac{-3}{5} \times \frac{-3}{5}$
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Sign up nowWhich of the following is not reciprocal of $\left(\frac{2}{3}\right)^{4}$ ?
(a) $\left(\frac{3}{2}\right)^{4}$
(b) $\left(\frac{2}{3}\right)^{-4}$
(c) $\left(\frac{3}{2}\right)^{-4}$
(d) $\frac{3^{4}}{2^{4}}$
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Sign up nowWhich of the following numbers is not equal to $\frac{-8}{27}$ ?
(a) $\left(\frac{2}{3}\right)^{-3}$
(b) $-\left(\frac{2}{3}\right)^{3}$
(c) $\left(-\frac{2}{3}\right)^{3}$
(d) $\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)$
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Sign up now$\left(\frac{2}{3}\right)^{-5}$ is equal to
(a) $\left(\frac{-2}{3}\right)^{5}$
(b) $\left(\frac{3}{2}\right)^{5}$
(c) $\frac{2 x-5}{3}$
(d) $\frac{2}{3 \times 5}$
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Sign up now$\left(\frac{-1}{2}\right)^{5} \times\left(\frac{-1}{2}\right)^{3}$ is equal to
(a) $\left(\frac{-1}{2}\right)^{8}$
(b) $-\left(\frac{1}{2}\right)^{8}$
(c) $\left(\frac{1}{4}\right)^{8}$
(d) $\left(-\frac{1}{2}\right)^{15}$
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Sign up now$\left(-\frac{1}{5}\right)^{5}$
(b) $\left(-\frac{1}{5}\right)^{11}$
(c) $(-5)^{5}$
(d) $\left(\frac{1}{5}\right)^{5}$
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Sign up now$\left(\frac{-2}{5}\right)^{7} \div\left(\frac{-2}{5}\right)^{5}$ is equal to
(a) $\frac{4}{25}$
(b) $\frac{-4}{25}$
(c) $\left(\frac{-2}{5}\right)^{12}$
(d) $\frac{25}{4}$
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Sign up now$\left\{\left(\frac{1}{3}\right)^{2}\right\}^{4}$ is equal to
(a) $\left(\frac{1}{3}\right)^{6}$
(b) $\left(\frac{1}{3}\right)^{8}$
(c) $\left(\frac{1}{3}\right)^{24}$
(d) $\left(\frac{1}{3}\right)^{16}$
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Sign up now$\left(\frac{1}{5}\right)^{0}$ is equal to
(a) 0
(b) $\frac{1}{5}$
(c) 1
(d) 5
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Sign up now$\left(\frac{-3}{2}\right)^{-1}$ is equal to
(a) $\frac{2}{3}$
(b) $-\frac{2}{3}$
(c) $\frac{3}{2}$
(d) none of these
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Sign up now$\left(\frac{2}{3}\right)^{-5} \times\left(\frac{5}{7}\right)^{-5}$ is equal to
(a) $\left(\frac{2}{3} \times \frac{5}{7}\right)^{-10}$
(b) $\left(\frac{2}{3} \times \frac{5}{7}\right)^{-5}$
(c) $\left(\frac{2}{3} \times \frac{5}{7}\right)^{25}$
(d) $\left(\frac{2}{3} \times \frac{5}{7}\right)^{-25}$
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Sign up now$\left(\frac{3}{4}\right)^{5} \div\left(\frac{5}{3}\right)^{5}$ is equal to
(a) $\left(\frac{3}{4} \div \frac{5}{3}\right)^{5}$
(b) $\left(\frac{3}{4} \div \frac{5}{3}\right)^{1}$
(c) $\left(\frac{3}{4} \div \frac{5}{3}\right)^{0}$
(d) $\left(\frac{3}{4} \div \frac{5}{3}\right)^{10}$
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Sign up nowFor any two non-zero rational numbers $a$ and $b, a^{4} \div b^{4}$ is equal to
(a) $(a \div b)^{1}$
(b) $(a \div b)^{0}$
(c) $(a \div b)^{4}$
(d) $(a \div b)^{8}$
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Sign up nowFor any two rational numbers $a$ and $b, a^{5} \times b^{5}$ is equal to
(a) $(a \times b)^{0}$
(b) $(a \times b)^{10}$
(c) $(a \times b)^{5}$
(d) $(a \times b)^{25}$
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Sign up nowFor a non-zero rational number $a, a^{7} \div a^{12}$ is equal to
(a) $a^{5}$
(b) $a^{-19}$
(c) $a^{-5}$
(d) $a^{19}$
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Sign up nowFor a non zero rational number $a,\left(a^{3}\right)^{-2}$ is equal to
(a) $a^{6}$
(b) $a^{-6}$
(c) $a^{-9}$
(d) $a^{1}$
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