Question

How do you simplify the given term ${{\left( 4\times {{10}^{-5}} \right)}^{-6}}$?

Answer 1

Hint: We start solving the problem by equating the given term to a variable. We then make use of the facts that $4={{2}^{2}}$, $10=2\times 5$ to proceed through the problem. We then make use of laws of exponents ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$, ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ to proceed further through the problem. We then make the necessary calculations to get the required answer for the given problem.

Complete step-by-step answer:
According to the problem, we are asked to simplify the given term ${{\left( 4\times {{10}^{-5}} \right)}^{-6}}$.
Let us assume $t={{\left( 4\times {{10}^{-5}} \right)}^{-6}}$ ---(1).
We know that $4={{2}^{2}}$, $10=2\times 5$. Let us use these results in equation (1).
$\Rightarrow t={{\left( {{2}^{2}}\times {{\left( 2\times 5 \right)}^{-5}} \right)}^{-6}}$ ---(2).
From the laws of exponents, we know that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. Let us use this result in equation (2).
$\Rightarrow t={{\left( {{2}^{2}}\times {{2}^{-5}}\times {{5}^{-5}} \right)}^{-6}}$ ---(3).
From the laws of exponents, we know that ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, for $m\ge n$. Let us use this result in equation (3).
$\Rightarrow t={{\left( {{2}^{2-5}}\times {{5}^{-5}} \right)}^{-6}}$.
$\Rightarrow t={{\left( {{2}^{-3}}\times {{5}^{-5}} \right)}^{-6}}$ ---(4).
From the law of exponents, we know that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. Let us use this result in equation (5).
$\Rightarrow t={{2}^{-3\times -6}}\times {{5}^{-5\times -6}}$.
$\Rightarrow t={{2}^{18}}\times {{5}^{30}}$.
So, we have found the simplified form of the term ${{\left( 4\times {{10}^{-5}} \right)}^{-6}}$ as ${{2}^{18}}\times {{5}^{30}}$.
$\therefore $ The simplified form of the term ${{\left( 4\times {{10}^{-5}} \right)}^{-6}}$ is ${{2}^{18}}\times {{5}^{30}}$.

Note: We should perform each step carefully to avoid confusion and calculation mistakes. We can also solve the given problem as shown below:
We have $t={{\left( 4\times {{10}^{-5}} \right)}^{-6}}$ ---(5).
From the laws of exponents, we know that ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. Let us use this result in equation (5).
$\Rightarrow t={{\left( \dfrac{4}{{{10}^{5}}} \right)}^{-6}}$ ---(7).
We know that $4={{2}^{2}}$, $10=2\times 5$. Let us use these results in equation (7).
$\Rightarrow t={{\left( \dfrac{{{2}^{2}}}{{{\left( 2\times 5 \right)}^{5}}} \right)}^{-6}}$ ---(8).
From the law of exponents, we know that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. Let us use this result in equation (8).
$\Rightarrow t={{\left( \dfrac{{{2}^{2}}}{{{2}^{5}}\times {{5}^{5}}} \right)}^{-6}}$ ---(9).
From the law of exponents, we know that $\dfrac{{{a}^{m}}}{{{a}^{n}}}=\dfrac{1}{{{a}^{n-m}}}$, for $m$\Rightarrow t={{\left( \dfrac{1}{{{2}^{5-2}}\times {{5}^{5}}} \right)}^{-6}}$.
$\Rightarrow t={{\left( \dfrac{1}{{{2}^{3}}\times {{5}^{5}}} \right)}^{-6}}$ ---(10).
From the laws of exponents, we know that ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. Let us use this result in equation (10).
$\Rightarrow t={{\left( {{2}^{3}}\times {{5}^{5}} \right)}^{6}}$ ---(11).
From the law of exponents, we know that ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. Let us use this result in equation (11).
$\Rightarrow t={{2}^{3\times 6}}\times {{5}^{5\times 6}}$.
$\Rightarrow t={{2}^{18}}\times {{5}^{30}}$.