CAT 2019

Slot 1

QA

Question & Solution

Step 1: Express all the terms in terms of prime factors.

We are given the equation: $$ (\sqrt{2})^{19}\cdot 3^4\cdot 4^2\cdot 9^m\cdot 8^n = 3^n \cdot 16^m \cdot (\sqrt[4]{64}) $$

Let’s break down each term into prime factors:

• $(\sqrt{2})^{19} = 2^{19/2}$ (since $\sqrt{2} = 2^{1/2}$)
• $3^4$ remains as it is.
• $4^2 = (2^2)^2 = 2^4$
• $9^m = (3^2)^m = 3^{2m}$
• $8^n = (2^3)^n = 2^{3n}$
• $3^n$ remains as it is.
• $16^m = (2^4)^m = 2^{4m}$
• $\sqrt[4]{64} = \sqrt[4]{2^6} = 2^{6/4} = 2^{3/2}$

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Step 2: Substitute these values back into the equation.

Substitute the prime factorized forms into the given equation: $$ 2^{19/2}\cdot 3^4\cdot 2^4 \cdot 3^{2m}\cdot 2^{3n} = 3^n \cdot 2^{4m}\cdot 2^{3/2} $$

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Step 3: Simplify the equation.

Now combine the powers of 2 and 3 on both sides.

• On the left-hand side, combine the powers of 2: $$ 2^{19/2 + 4 + 3n} \times 3^{4 + 2m} $$

• On the right-hand side, combine the powers of 2: $$ 3^n \times 2^{4m + 3/2} $$

Thus, the equation becomes: $$ 2^{19/2 + 4 + 3n} \times 3^{4 + 2m} = 3^n \times 2^{4m + 3/2} $$

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Step 4: Equate the powers of 2 and 3.

For the equation to hold, the powers of 2 and 3 on both sides must be equal.

• For the powers of 3: $$ 4 + 2m = n $$

• For the powers of 2: $$ \frac{19}{2} + 4 + 3n = 4m + \frac{3}{2} $$

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Step 5: Solve the system of equations.

From the first equation ( 4 + 2m = n ), solve for ( n ): $$ n = 4 + 2m $$

Substitute this value of ( n ) into the second equation: $$ \frac{19}{2} + 4 + 3(4 + 2m) = 4m + \frac{3}{2} $$

Simplify the equation: $$ \frac{19}{2} + 4 + 12 + 6m = 4m + \frac{3}{2} $$

Multiply through by 2 to eliminate the fractions: $$ 19 + 8 + 24 + 12m = 8m + 3 $$

Simplify further: $$ 51 + 12m = 8m + 3 $$

Solve for ( m ): $$ 51 - 3 = 8m - 12m $$ $$ 48 = -4m $$ $$ m = -12 $$

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Final Answer: Thus, the value of ( m ) is: $ \boxed{-12} $