The value of {[(2/3)²]³× (1/3)^{-4}+3^{-1}× 1/6}{(5/8)^{-3}× (5/4)²× 2^{-1}} is:

Correct Answer is : 21596215/9696215

Correct Answer is : 21596215/9696215

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CTET Paper II (Mathematics and Science) May 2016 Solved Paper

Question : 59 of 150

Marks: +1, -0

The value of

\frac{[(\frac{2}{3})^{2}]^{3}\times (\frac{1}{3})^{-4}+3^{-1}\times \frac{1}{6}}{(\frac{5}{8})^{-3}\times (\frac{5}{4})^{2}\times 2^{-1}}

is:

Solution:

Concept:

Use laws of exponents:

(a^m)^n = a^{mn}

a^{-n} = 1/a^n

, and

a^m \times a^n = a^{m+n}

. Simplify numerator and denominator separately before dividing.

Explanation:

First, simplify the numerator:

[(\frac{2}{3})^{2}]^{3} = (\frac{2}{3})^{6} = \frac{2^{6}}{3^{6}}

Multiply by

(\frac{1}{3})^{-4} = 3^{4}

gives

\frac{2^{6}}{3^{6}} \times 3^{4} = \frac{2^{6}}{3^{2}} = \frac{64}{9}

Add

3^{-1}\times \frac{1}{6} = \frac{1}{3} \times \frac{1}{6} = \frac{1}{18}

Numerator becomes

\frac{64}{9} + \frac{1}{18} = \frac{128}{18} + \frac{1}{18} = \frac{129}{18}

Now simplify the denominator:

(\frac{5}{8})^{-3} = (\frac{8}{5})^{3} = \frac{8^{3}}{5^{3}} = \frac{512}{125}

Multiply by

(\frac{5}{4})^{2} = \frac{25}{16}

gives

\frac{512}{125} \times \frac{25}{16} = \frac{512 \times 25}{125 \times 16} = \frac{32 \times 25}{125} = \frac{800}{125} = \frac{32}{5}

Then multiply by

2^{-1} = \frac{1}{2}

\frac{32}{5} \times \frac{1}{2} = \frac{16}{5}

Now the expression is

\frac{129/18}{16/5} = \frac{129}{18} \times \frac{5}{16} = \frac{129 \times 5}{18 \times 16} = \frac{645}{288}

Simplify by dividing numerator and denominator by 3:

\frac{645 \div 3}{288 \div 3} = \frac{215}{96}

Answer:

\frac{215}{96}

(Option B)

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