Question

Find the value of the expression, \[\dfrac{{0.85 \times 0.85 \times 0.85 + 0.15 \times 0.15 \times 0.15}}{{0.85 \times 0.85 - 0.85 \times 0.15 + 0.15 \times 0.15}}\].

Answer 1

Hint: Let us take \[a = 0.85\] and \[b = 0.15\] in the given expression. Then use the property of sum of cube of two numbers, that is, \[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\] to simplify the given expression.
Apply this property, and then use the given conditions to find the required value.

Complete step-by-step solution:

It is given that the expression is \[\dfrac{{0.85 \times 0.85 \times 0.85 + 0.15 \times 0.15 \times 0.15}}{{0.85 \times 0.85 - 0.85 \times 0.15 + 0.15 \times 0.15}}\].

Simplifying the above expression to find the required value, we get

\[\dfrac{{{{\left( {0.85} \right)}^3} + {{\left( {0.15} \right)}^3}}}{{{{\left( {0.3} \right)}^2} - 0.85 \times 0.15 + {{\left( {0.15} \right)}^2}}}\]

Let us take \[a = 0.85\] and \[b = 0.15\] in the above expression to simplify the above expression.

Using these values in the above expression, we get

\[\dfrac{{{a^3} + {b^3}}}{{{a^2} - ab + {b^2}}}\]

Multiplying the numerator and denominator by \[a + b\] in the above expression, we get

 \[\dfrac{{\left( {a + b} \right)\left( {{a^3} + {b^3}} \right)}}{{\left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}}\]

We know the property of sum of cube of two numbers, that is, \[{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\].

Using the above property of cube in the denominator of the above expression, we get

\[ \Rightarrow \dfrac{{\left( {a + b} \right)\left( {{a^3} + {b^3}} \right)}}{{{a^3} + {b^3}}}\]

Simplifying the above expression, we get

\[ \Rightarrow a + b\]

Replacing \[0.85\] for \[a\] and \[0.15\] for \[b\] in the above expression, we get

\[
   \Rightarrow 0.85 + 0.15 \\
   \Rightarrow 1 \\
\]

Thus, the required value of the given expression is 1.

Note: In solving these types of questions, you should be familiar with the property of the cube. Some students try to find the value by solving the given expression, which is really time consuming. Then use the given conditions and values given in the question, and substitute the values in this property, to find the required value. Also, we are supposed to write the values properly to avoid any miscalculation.