Question

How do you get rid of the negative exponents in ${\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}}$?

Answer 1

Hint: Here we need to eliminate the negative exponents in the given expression. We will use the different properties of the exponential function. We will first use the distributive property of exponents to distribute the exponents for the terms inside the parenthesis. Then we will use the negative exponent’s rule to get the final answer in which there is no term with the negative power.

Formula used:
We will use the following formulas:
1. Distributive property of exponents:- ${\left( {a \times b} \right)^c} = {a^c} \times {b^c}$
2. Negative power rule of exponents:- ${a^{ - b}} = \dfrac{1}{{{a^b}}}$

Complete step by step solution:
Here we need to eliminate the negative exponents in the given expression and the given expression is
${\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}}$.
We know from the distributive property of exponents that:
${\left( {a \times b} \right)^c} = {a^c} \times {b^c}$
We will use the same property here in the given expression.
${\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}} = {\left( { - 4} \right)^{ - 2}}{a^{3 \times - 2}}{b^{ - 5}}^{ \times - 2}$
On multiplying the exponents, we get
$ {\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}}= {\left( { - 4} \right)^{ - 2}}{a^{ - 6}}{b^{10}}$
We know from the Negative power rule of exponents that: ${a^{ - b}} = \dfrac{1}{{{a^b}}}$.
We will use the negative power rule of exponents here for the terms having negative exponents. Therefore, we get
$ {\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}} = \dfrac{1}{{{{\left( { - 4} \right)}^2}}} \times \dfrac{1}{{{a^6}}} \times {b^{10}}$
Now, we will apply the exponents on the bases
$ {\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}}= \dfrac{1}{{16}} \times \dfrac{1}{{{a^6}}} \times {b^{10}}$
On multiplying the terms, we get
$ {\left( { - 4{a^3}{b^{ - 5}}} \right)^{ - 2}} = \dfrac{{{b^{10}}}}{{16 \cdot {a^6}}}$

Hence, this is a simplified expression in which no term is having the negative exponent.

Note:
Here we have obtained the simplified expression in which there is no term that has negative exponents. The exponent is defined as the number, which is positioned at the up-right of the base number. We need to remember the basic rules of the exponents. Remember that if we multiply the exponents having the same bases, then their exponents get added. Similarly when we divide the exponentials having the same bases, then their exponents get subtracted.