Question

Write with a single exponent, 9^2(12^3)(2)

Answer 1

`6^7`

Explanation:

`9^2(12^3)(2)`

`\mathsf{rewrite \ 9 \ as \ 3^2 } \implies 9^2=(3^2)^2`

Apply exponent rule
`(a^b)^c=a^(bc)`:

`\implies (3^2)^2=3^4`

`\mathsf{rewrite \ 12 \ as \ 2 \cdot 2 \cdot 3 } \implies 12^3=(2 \cdot 2 \cdot 3)^3`

Apply exponent rule
`(a\cdot b)^c=a^c \cdot b^c`

`\implies (2 \cdot 2 \cdot 3)^3=2^3 \cdot 2^3 \cdot 3^3`

`\mathsf{rewrite \ 2 \ as \ 2^1}`

Therefore,

`9^2(12^3)(2)=3^4 \cdot2^3\cdot2^3\cdot3^3\cdot2^1`

Gather like terms

`\implies 3^4\cdot3^3 \cdot2^3\cdot2^3\cdot2^1`

Apply exponent rule
`a^b \cdot a^c=a^(b+c)`:

`\implies 3^(4+3)\cdot2^(3+3+1)`

`\implies 3^7 \cdot 2^7`

Apply exponent rule
`a^c \cdot b^c=(ab)^c`

`\implies (3 \cdot2)^7`

`\implies 6^7`

Answer 2

9^2(12^3)(2)
81(1728)(2)
81(3456)
279,936

279,936 is equivalent to 6^7

Hope this helps!