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How do you simplify ${\left( {\dfrac{2}{5}} \right)^{ - 2}}$ and write it using only positive exponents?

Last updated date: 22nd Jul 2024
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Hint: To simplify a fraction with negative exponents and represent it with positive exponents, one can simply take the reciprocal of the fraction and remove the negative sign from the exponent. For further simplification, multiply the numerator and denominator with itself as the exponent is 2.

Step by step solution:
To simplify the fraction, we need to convert the negative exponent to positive.
As we know that a negative exponent can be written as positive by writing it in the denominator with numerator as 1 i.e., ${a^{ - b}} = \dfrac{1}{{{a^b}}}$
Applying the above formula here,
$\Rightarrow {\left( {\dfrac{2}{5}} \right)^{ - 2}} = \dfrac{1}{{{{\left( {\dfrac{2}{5}} \right)}^2}}}$
Now, opening the parenthesis on the right-hand side and writing the exponent separately with numerator and denominator,
$\Rightarrow {\left( {\dfrac{2}{5}} \right)^{ - 2}} = \dfrac{1}{{\dfrac{{{2^2}}}{{{5^2}}}}}$
Now, we know that a denominator’s denominator can be written in the numerator. Applying the rule here, we get
$\Rightarrow {\left( {\dfrac{2}{5}} \right)^{ - 2}} = \dfrac{{{5^2}}}{{{2^2}}}$
This is the simplified version with positive exponents
Now, simplifying further and multiplying the numerator and denominator with itself as the exponent is 2
$\Rightarrow {\left( {\dfrac{2}{5}} \right)^{ - 2}} = \dfrac{{5 \times 5}}{{2 \times 2}}$
$\Rightarrow {\left( {\dfrac{2}{5}} \right)^{ - 2}} = \dfrac{{25}}{4}$

Thus, the final answer after simplification is $\dfrac{{25}}{4}$.

It is important to note that a fraction with a negative exponent is just a reciprocal of the fraction with a positive exponent. Also, the exponent means a repeated multiplication of the base. As the exponent here is 2, we will multiply the fraction twice. If the exponent was 3, we multiply the fraction thrice.