Depth · one concept from first principles · step 2 of 4

Solve for the building blocks, then rebuild

If $a=3^{x+2}$ and $b=2^{x+1}$ , express $12^{3x}$ in terms of $a$ and $b$ . (You may use the fact that $6^6=46656$ .)

Expressions

Your answer

How to solve

Strategy Convert to Algebra — 12 is built from 2s and 3s, so 12^(3x) is built from 2^x and 3^x. First express those two building blocks with a and b, then assemble 12^(3x) from them.

1STEP 1

Express 3^x with a

Split off the +2 in the exponent.

a=3^(x+2)=3^x×3²=9×3^x → 3^x=a/9

2STEP 2

Express 2^x with b

Split off the +1 in the exponent.

b=2^(x+1)=2×2^x → 2^x=b/2

3STEP 3

Break 12^(3x) into the building blocks

Use 12=2²×3, then group the exponents around x.

12^(3x)=(2²×3)^(3x)=2^(6x)×3^(3x)=(2^x)⁶×(3^x)³

4STEP 4

Substitute

Replace 2^x and 3^x with the expressions from the first two steps.

(b/2)⁶×(a/9)³=b⁶/64×a³/729

5STEP 5

Combine the denominators

64×729=2⁶×3⁶=6⁶, which the problem says is 46656.

a³b⁶/(2⁶×3⁶)=a³b⁶/(6⁶)=a³b⁶/46656

Answer

a³b⁶/46656

Test x=1: a=27, b=4, and 12³=1728. The formula gives (27³×4⁶)/46656=(19683×4096)/46656=80621568/46656=1728. It matches.

Takeaway

Find the small pieces (2^x, 3^x) in the given letters first; then any power made of them follows.

Express 3^x with a
Express 2^x with b
Break 12^(3x) into the building blocks
Substitute
Combine the denominators

Where next?

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