CAT 2019 Surds and Indices Question (Hard)

Question

If and are integers such that cdot 3 4 4 2 cdot 9 m cdot 8 n 16 m cdot sqrt 4 64 Then what is the value of ?

Accepted Answer

Step 1 Express all terms as powers of primes - - - - - --- Step 2 Rewrite both sides with simplified bases LHS becomes cdot 3 4 2 4 cdot 3 2m cdot 2 3n 2 19 - 4 - 3n cdot 3 4 - 2m RHS becomes 2 4m cdot 2 3 2 3 n cdot 2 -4m - 3 2 --- Step 3 Equating powers of 2 and 3 For base 2 19 - 4 - 3n -4m - 2 Ri

CAT 2019

Quantitative

Surds and Indices

Hard

If $m$ and $n$ are integers such that:
$\frac{2^{19} \cdot 3^4}{4^2 \cdot 9^m \cdot 8^n} = \frac{3^n}{16^m \cdot \sqrt[4]{64}}$
Then what is the value of $m$ ?

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