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$\dfrac{0.83\times 0.83\times 0.83+0.17\times 0.17\times 0.17}{0.83\times 0.83-0.83\times 0.17+0.17\times 0.17}$ is equal to
$\begin{align}
  & A)1 \\
 & B){{\left( 0-83 \right)}^{3}}+{{(0.17)}^{3}} \\
 & C)0 \\
 & D)\text{None of these} \\
\end{align}$

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Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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Hint: In this problem, we have to simplify the given algebraic expression to get the answer. So, we will use the algebraic identity to get the solution. So, we will first simplify the expression by using the formula ${{a}^{2}}=a\times a$ and ${{b}^{3}}=b\times b\times b$ in the expression. Then, on further solving, we will apply the algebraic identity ${{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}-ab+{{b}^{2}})$ on the numerator in the algebraic expression. As we know, the same terms in the division cancel out each other with the quotient 1 and remainder 0, thus we will make the necessary calculations, to get the required result for the solution.

Complete step by step solution:
According to the question, we have to simplify an algebraic expression.
Thus, we will use the algebraic identity to get the solution.
The algebraic expression given to us is $\dfrac{0.83\times 0.83\times 0.83+0.17\times 0.17\times 0.17}{0.83\times 0.83-0.83\times 0.17+0.17\times 0.17}$ ---- (1)
First, we will simplify the expression (1) by using the formula ${{a}^{2}}=a\times a$ and ${{b}^{3}}=b\times b\times b$ , we get
$\dfrac{{{\left( 0.83 \right)}^{3}}+{{\left( 0.17 \right)}^{3}}}{{{\left( 0.83 \right)}^{2}}-0.83\times 0.17+{{\left( 0.17 \right)}^{2}}}$
Now, we will apply the algebraic identity ${{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}-ab+{{b}^{2}})$ on the numerator in the above algebraic expression, we get
$\dfrac{\left( 0.83+0.17 \right)\left( {{\left( 0.83 \right)}^{2}}-0.83\times 0.17+{{\left( 0.17 \right)}^{2}} \right)}{{{\left( 0.83 \right)}^{2}}-0.83\times 0.17+{{\left( 0.17 \right)}^{2}}}$
Now, we know that the same terms in the division cancel out each other with the quotient 1 and remainder 0, therefore we get
$\left( 0.83+0.17 \right)$
Now, on adding the numbers of the above expression, we get
$1$ which is the required solution.

So, the correct answer is “Option A”.

Note: While solving this problem, do step-by-step calculations to avoid confusion and mathematical errors. You can also solve this problem by simply using the basic mathematical rules but that makes your answer quite long, so we use the algebraic identity.