Question

Find the value of ${{\left( 25 \right)}^{3}}$ using the short-cut method.

Answer 1

Hint: We will write the number 25 as a sum of two numbers. Then we will use the algebraic identity to expand the cube of the sum of two numbers. This identity is ${{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}$. We will find the cube of $a$ and $b$ by simple multiplications. After adding all the terms from the identity, we will get the cube of 25.

Complete step-by-step solution
Let us write $25=20+5$. So, we can write ${{\left( 25 \right)}^{3}}={{\left( 20+5 \right)}^{3}}$. Now, we have an algebraic identity which tells us how to expand the cube of a sum of two numbers. The identity is ${{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}+{{b}^{3}}$. We have $a=20$ and $b=5$. Substituting these values in the identity, we get
${{\left( 20+5 \right)}^{3}}={{\left( 20 \right)}^{3}}+3{{\left( 20 \right)}^{2}}\left( 5 \right)+3\left( 20 \right){{\left( 5 \right)}^{2}}+{{\left( 5 \right)}^{3}}$
Now, we know that ${{2}^{3}}=8$ and hence, ${{\left( 20 \right)}^{3}}=8000$. Also, we know that ${{2}^{2}}=4$ which tells us that ${{\left( 20 \right)}^{2}}=400$. We know that ${{5}^{2}}=25$ and ${{5}^{3}}=125$. Substituting all these values in the above expression, we get the following
${{\left( 20+5 \right)}^{3}}=8000+3\times 400\times 5+3\times 20\times 25+125$
Simplifying this equation, we get
$\begin{align}
  & {{\left( 20+5 \right)}^{3}}=8000+6000+1500+125 \\
 & \therefore {{\left( 25 \right)}^{3}}=15625 \\
\end{align}$
Hence, we get the cube of 25 as 15625.

Note: We can use the algebraic identity which expands the cube of the difference of two numbers as well. This identity is ${{\left( a-b \right)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}$. We should split up the two-digit number according to convenience and then use one of the two algebraic identities to expand the cube. For example, if we have a number 39, then it is convenient for calculations to write it as $39=40-1$ instead of $39=30+9$. In the case of 25, it is a matter of preference to split it as a sum of two numbers or as a difference of two numbers.