Value of ( a1/8 + a-1/8 ) ( a1/8 - a-1/8 ) ( a1/4 + a-1/4 ) ( a1/2 + a-1/2 ) is:

Option b: a- 1/a. Solution: We are given the expression: E = ( a1/8 + a-1/8 ) ( a1/8 - a-1/8 ) ( a1/4 + a-1/4 ) ( a1/2 + a-1/2 ) Let us simplify the expression step-by-step using the difference of squares identity, (u+v)(u-v) = u2 - v2 : 1) Combine the first two terms: ( a1/8 + a-1/8 ) ( a1/8 - a-1/8 ) = ( a1/8 )2 - ( a-1/8 )2 = a1/4 - a-1/4 2) Now, multiply this result by the third term: ( a1/4 - a-1/4 ) ( a1/4 + a-1/4 ) = ( a1/4 )2 - ( a-1/4 )2 = a1/2 - a-1/2 3) Finally, multiply this result by the fourth term: ( a1/2 - a-1/2 ) ( a1/2 + a-1/2 ) = ( a1/2 )2 - ( a-1/2 )2 = a1 - a-1 = a - 1/a Hence, Option B is the correct answer.

Ratio, Proportion, Indices, LogarithmMCQMTP June 23 Series IIQuestion 919 of 305

All Questions

Value of $\displaystyle \left( a^{1/8} + a^{-1/8} \right) \left( a^{1/8} - a^{-1/8} \right) \left( a^{1/4} + a^{-1/4} \right) \left( a^{1/2} + a^{-1/2} \right)$ is:

Options

\displaystyle a+ \frac{1}{a}

\displaystyle a- \frac{1}{a}

\displaystyle a^2 + \frac{1}{a^2}

\displaystyle a^2 - \frac{1}{a^2}

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Correct Answer

✅ Option b — $\displaystyle a- \frac{1}{a}$

All Options:

A $\displaystyle a+ \frac{1}{a}$
B $\displaystyle a- \frac{1}{a}$
C $\displaystyle a^2 + \frac{1}{a^2}$
D $\displaystyle a^2 - \frac{1}{a^2}$

Detailed Solution & Explanation

We are given the expression:

E = \left( a^{1/8} + a^{-1/8} \right) \left( a^{1/8} - a^{-1/8} \right) \left( a^{1/4} + a^{-1/4} \right) \left( a^{1/2} + a^{-1/2} \right)

Let us simplify the expression step-by-step using the difference of squares identity,

\displaystyle (u+v)(u-v) = u^2 - v^2

:

1) Combine the first two terms:

\left( a^{1/8} + a^{-1/8} \right) \left( a^{1/8} - a^{-1/8} \right) = \left( a^{1/8} \right)^2 - \left( a^{-1/8} \right)^2 = a^{1/4} - a^{-1/4}

2) Now, multiply this result by the third term:

\left( a^{1/4} - a^{-1/4} \right) \left( a^{1/4} + a^{-1/4} \right) = \left( a^{1/4} \right)^2 - \left( a^{-1/4} \right)^2 = a^{1/2} - a^{-1/2}

3) Finally, multiply this result by the fourth term:

\left( a^{1/2} - a^{-1/2} \right) \left( a^{1/2} + a^{-1/2} \right) = \left( a^{1/2} \right)^2 - \left( a^{-1/2} \right)^2 = a^1 - a^{-1} = a - \frac{1}{a}

Hence, **Option B** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

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Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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Value of (a1/8+a−1/8)(a1/8−a−1/8)(a1/4+a−1/4)(a1/2+a−1/2)\displaystyle \left( a^{1/8} + a^{-1/8} \right) \left( a^{1/8} - a^{-1/8} \right) \left( a^{1/4} + a^{-1/4} \right) \left( a^{1/2} + a^{-1/2} \right)(a1/8+a−1/8)(a1/8−a−1/8)(a1/4+a−1/4)(a1/2+a−1/2) is: