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

Answer:

$n=30$

Step-by-step explanation:

Given expression:

$3^1\times 9^2\times 27^3\times 81^4$

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

Applying the exponents rule to simplify.

$3^1\times 9^2\times 27^3\times 81^4$

Writing all numbers as product of 3.

⇒ $3^1\times (3\times3)^2\times (3\times3\times3)^3\times (3\times3\times3\times3)^4$

Writing each in exponents form.

⇒ $3^1\times (3^2)^2\times (3^3)^3\times (3^4)^4$

Using power of a power rule $[\ (a^x)^y=a^{xy}\ ]$

⇒ $3^1\times 3^4\times 3^9\times 3^16$

Using multiplication rule of exponents $[\ a^x\times a^y=a^{x+y}\ ]$

⇒ $3^{1+4+9+16}$

⇒ $3^{30}$

So we have $3^n=3^{30}$

∴ $n=30$