Question

Evaluate $\left[ {{\left( \dfrac{1}{3} \right)}^{-3}}-{{\left( \dfrac{1}{2} \right)}^{-3}} \right]\div {{\left( \dfrac{1}{4} \right)}^{-3}}$

Answer 1

Hint: To solve the above expression I need to know the concept of power and exponent. The concept required to solve the problem is that when a number having negative power is changed to positive by interchanging the numerator and the denominator of the fraction. Mathematically it will written as ${{\left( a \right)}^{-n}}={{\left( \dfrac{1}{a} \right)}^{n}}$.

Complete step-by-step solution:
The question ask us to find $\left[ {{\left( \dfrac{1}{3} \right)}^{-3}}-{{\left( \dfrac{1}{2} \right)}^{-3}} \right]\div {{\left( \dfrac{1}{4} \right)}^{-3}}$. The first step is to change the sign of the power. We will change the sign of power of the number in the bracket. On applying the formula ${{\left( \dfrac{1}{a} \right)}^{-n}}={{\left( a \right)}^{n}}$ we get ${{\left( \dfrac{1}{3} \right)}^{-3}}={{\left( 3 \right)}^{3}}$,\[{{\left( \dfrac{1}{2} \right)}^{-3}}={{\left( 2 \right)}^{3}}\] and ${{\left( \dfrac{1}{4} \right)}^{-3}}={{\left( 4 \right)}^{3}}$. On applying the above formula in the question we get:
$\Rightarrow \left[ {{\left( \dfrac{1}{3} \right)}^{-3}}-{{\left( \dfrac{1}{2} \right)}^{-3}} \right]\div {{\left( \dfrac{1}{4} \right)}^{-3}}$
$\Rightarrow \left[ {{\left( 3 \right)}^{3}}-{{2}^{3}} \right]\div {{\left( 4 \right)}^{3}}$
The second step is to apply the BODMAS rule in the above expression. On applying the rule we will solve the numbers inside the bracket. For this we will expand the terms in the formula ${{a}^{n}}=a\times a\times a.......ntimes$. In the given question value of $n$ is $3$.
$\Rightarrow \left[ 3\times 3\times 3-2\times 2\times 2 \right]\div \left( 4\times 4\times 4 \right)$
On multiplying the terms we get:
$\Rightarrow \left[ 27-8 \right]\div 64$
$\Rightarrow 19\div 64$
The term which is being divided is multiplied to the reciprocal as given below:
$\Rightarrow 19\times \dfrac{1}{64}$
$\Rightarrow \dfrac{19}{64}$
Since there is no common factor in the numerator and the denominator of the fraction, the fraction cannot be changed further in the lowest term.
On writing the above form of fraction in decimal form we get:
$\Rightarrow 0.296$
$\therefore $ The expression $\left[ {{\left( \dfrac{1}{3} \right)}^{-3}}-{{\left( \dfrac{1}{2} \right)}^{-3}} \right]\div {{\left( \dfrac{1}{4} \right)}^{-3}}$ results in $\dfrac{19}{64}$ which on decimal form is $0.296$.

Note: Always remember to use the BODMAS rule which says that in the expression given the operation used will be in the series of Bracket, Of, Division, Multiplication, Addition and Subtraction. The formula ${{a}^{n}}$ on expansion will be $a\times a\times a\times a.......n-times$.