Question

If the expression $3^1 times 9^2 times 27^3 \times 81^4 is simplified to the form 3^n, what is the value of n?

Answer 1

`n=30`

Explanation:

Given expression:

`3^1* 9^2* 27^3* 81^4`

The given expression needs to be simplified to the form
`3^n`

Applying the exponents rule to simplify.

`3^1* 9^2* 27^3* 81^4`

Writing all numbers as product of 3.

⇒
`3^1* (3*3)^2* (3*3*3)^3* (3*3*3*3)^4`

Writing each in exponents form.

⇒
`3^1* (3^2)^2* (3^3)^3* (3^4)^4`

Using power of a power rule
`[\ (a^x)^y=a^(xy)\ ]`

⇒
`3^1* 3^4* 3^9* 3^16`

Using multiplication rule of exponents
`[\ a^x* a^y=a^(x+y)\ ]`

⇒
`3^(1+4+9+16)`

⇒
`3^(30)`

So we have
`3^n=3^(30)`

∴
`n=30`