TheCATExam
Sign in

CAT 2019 Slot 1 — QA Question 4

IndicesEasy

If m and n are integers such that (√2)19 34 42 9m 8n = 3n 16m (64)1/4, then m is

Answer & solution

Correct answer: -12

  • A

    -16

  • -12

  • C

    -24

  • D

    -20

Solution

Easy

Everything is a power of 22 or 33. Rewrite every term in prime-base form, then equate the exponents of 22 and the exponents of 33 on the two sides. That gives two linear equations in m,nm,n; solve for mm.

1

Convert each base. Using 2=21/2\sqrt2=2^{1/2}, 4=224=2^2, 9=329=3^2, 8=238=2^3, 16=2416=2^4, 64=2664=2^6.

219/2⋅34⋅24⋅32m⋅23n=3n⋅24m⋅26/4\begin{aligned} &2^{19/2}\cdot 3^4\cdot 2^{4}\cdot 3^{2m}\cdot 2^{3n} = 3^{n}\cdot 2^{4m}\cdot 2^{6/4} \end{aligned}
2

Collect like bases. Combine the powers of 22 and of 33 on each side (64=32\tfrac{6}{4}=\tfrac32).

2 19/2+4+3n⋅3 4+2m=2 4m+3/2⋅3 n\begin{aligned} &2^{\,19/2+4+3n}\cdot 3^{\,4+2m} = 2^{\,4m+3/2}\cdot 3^{\,n} \end{aligned}
3

Equate exponents. Match powers of 22 and powers of 33.

192+4+3n=4m+32  ⇒  6n+24=8m(2)4+2m=n(3)\begin{aligned} &\tfrac{19}{2}+4+3n = 4m+\tfrac32 \;\Rightarrow\; 6n+24 = 8m \quad\text{(2)} \\ &4+2m = n \quad\text{(3)} \end{aligned}
4

Solve the system. Substitute n=4+2mn=4+2m from (3) into (2).

6(4+2m)+24=8m(sub step 3)⇒ 24+12m+24=8m⇒ 4m=−48⇒m=−12\begin{aligned} &6(4+2m)+24 = 8m \quad\text{(sub step 3)}\\ &\Rightarrow\ 24+12m+24 = 8m\\ &\Rightarrow\ 4m = -48 \Rightarrow m = -12 \end{aligned}
m=−12m = -12

Related Indices questions

See all Surds & Indices questions →
CAT 2019 Slot 1 QA Q4: If m and n are integers such that (√2) 19 3 4 4 2 9 m 8 n = 3 n 16 m (64) 1/4 , then m is — Solution | TheCATExam