Question

How do you use the laws of exponents to simplify the expression $\left( \dfrac{{{4}^{2}}}{{{4}^{3}}} \right)\times {{\left( 4 \right)}^{-3}}$?

Answer 1

Hint: From the question given we have to simplify the expression $\left( \dfrac{{{4}^{2}}}{{{4}^{3}}} \right)\times {{\left( 4 \right)}^{-3}}$ by using laws of exponents. To solve this question, we will use some laws of exponents they are $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{\left( m-n \right)}}$, ${{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$, ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. By using the above laws of exponents, we will be able to simplify the given expression.

Complete step by step solution:
From the question given we have to simplify the expression by using laws of exponents, the given expression is
$\Rightarrow \left( \dfrac{{{4}^{2}}}{{{4}^{3}}} \right)\times {{\left( 4 \right)}^{-3}}$
to simplify the above expression, we will use some laws of exponents like
$\Rightarrow \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{\left( m-n \right)}}$
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$
$\Rightarrow {{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$
Now the given expression is
$\Rightarrow \left( \dfrac{{{4}^{2}}}{{{4}^{3}}} \right)\times {{\left( 4 \right)}^{-3}}$
As we know that if any expression is any expression is in the form of $\dfrac{{{a}^{m}}}{{{a}^{n}}}$ by using laws of exponents it can be simplified as
$\Rightarrow \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{\left( m-n \right)}}$
So, in the given expression $\left( \dfrac{{{4}^{2}}}{{{4}^{3}}} \right)$ can be simplified as
$\Rightarrow {{\left( 4 \right)}^{\left( 2-3 \right)}}\times {{\left( 4 \right)}^{-3}}$
By further simplifying we will get,
$\Rightarrow {{\left( 4 \right)}^{-1}}\times {{\left( 4 \right)}^{-3}}$
As we know that if any expression is any expression is in the form of ${{a}^{m}}\times {{a}^{n}}$ by using laws of exponents it can be simplified as
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$
So, the expression will be simplified as
$\Rightarrow {{\left( 4 \right)}^{-1}}\times {{\left( 4 \right)}^{-3}}$
$\Rightarrow {{\left( 4 \right)}^{\left( -1+\left( -3 \right) \right)}}$
 By further simplifying we will get,
$\Rightarrow {{\left( 4 \right)}^{-4}}$
As we know that if any expression is any expression is in the form of ${{a}^{-m}}$ by using laws of exponents it can be written as
$\Rightarrow {{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$
 So, the expression can be written as
$\Rightarrow {{\left( 4 \right)}^{-4}}$
$\Rightarrow \dfrac{1}{{{\left( 4 \right)}^{4}}}$
By further simplifying we will get
$\Rightarrow \dfrac{1}{256}$

Note: Students should be very careful while doing the calculation part like in the step as students know that $ {{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$ but in our expression $ {{\left( 4 \right)}^{-1}}\times {{\left( 4 \right)}^{-3}}$ the powers are negative if students neglect the negative sign the whole answer will be wrong.