Question

Simplify: $\dfrac{{155 \times 155 \times 155 - 55 \times 55 \times 55}}{{155 \times 155 + 155 \times 55 + 55 \times 55}}$

Answer 1

Hint: Let $a = 155$ and $b = 55$ the numerator become ${a^3} - {b^3}$ and denominator become $\left( {{a^2} + ab + {b^2}} \right)$ . We know that ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$. Then denominator is cancelled out then answer become $a - b$

Complete step by step solution:
 $\dfrac{{155 \times 155 \times 155 - 55 \times 55 \times 55}}{{155 \times 155 + 155 \times 55 + 55 \times 55}}$
It also written as $\dfrac{{{{\left( {155} \right)}^3} - {{\left( {55} \right)}^3}}}{{{{\left( {155} \right)}^2} + 155 \times 55 + {{\left( {55} \right)}^2}}}$
We know that ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$ then
denominator is cancelled out and answer become

\[ \Rightarrow \left( {155 - 55} \right) = 100\]

Note: In these types of question find the term which can be cancelled out and has a form of identity as used in this question like ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$.