"25−32 by hand using properties of exponents." was answered by students like you

Mylo O'Moore

Answered question

2021-09-22

25^{- \frac{3}{2}}

by hand using properties of exponents.

Answer & Explanation

Aniqa O'Neill

Skilled2021-09-23Added 100 answers

Calculation:
$25^{- \frac{3}{2}}$ is in the form of $a^{- m}$ . Hence, using the second formula
$a^{- m} = \frac{1}{a^{m}}$ we rewrite $25^{- \frac{3}{2}}$ as $25^{- \frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}}$
In order to evaluate $\frac{1}{25^{\frac{3}{2}}}$ , first we evaluate $25^{\frac{3}{2}}$ which is in the denominator.

For this we rewrite $25^{\frac{3}{2}}$ as $25^{\frac{3}{2}} = 25^{(\frac{1}{2}) 3}$ .
Now using the first formula $a^{m n} = {(a^{m})}^{n} = {(a^{n})}^{m}$ , we rewrite $25^{(\frac{1}{2}) 3}$ as $25^{(\frac{1}{2}) 3} = (25^{\frac{1}{2}})^{3}$ .

We know that $25^{\frac{1}{2}} = \sqrt{25} = 5$ .
Hence, $25^{\frac{3}{2}} = 25^{(\frac{1}{2}) 3} = 5^{3} = 125$
As the denominator is evaluated, we can now substitute this in $25^{- \frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}}$ to complete the calculation.
Therefore, our original expression to be evaluated, $25^{- \frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}} = \frac{1}{125}$ i.e., $25^{- \frac{3}{2}} = \frac{1}{125}$
Conclusion:
Therefore, using the properties of exponent, $a^{m n} = {(a^{m})}^{n} = {(a^{n})}^{m}$ and $a^{- m} = \frac{1}{a^{m}}$ we easily found out $25^{- \frac{3}{2}} = \frac{1}{125}$ without using calculator.

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