Question

What value of n solves the equation?

3^n = 1/81

A.
–243

B.
–27

C.
–4

D.
4

Answer 1

Can you write 81 as a power of 3?

Answer 2

I honestly don't know, mathstudent55.

Answer 3

Could you tell me how?

Answer 4

\(3^1 = 3\)

\(3^2 = 3 \times 3 = 9\)

\(3^3 = 3 \times 3 \times 3 = 27\)

\(3^4 = 3 \times 3 \times 3 \times 3 = 81\)

Do you see that \(81 = 3^4\) ?

Answer 5

Yes.

Answer 6

I get it now.

Answer 7

Now look at the right side of the original problem.

You have \(\dfrac{1}{81} \)

Since we now know that \(81 = 3^4\), we can rewrite the right siuda as

\(\dfrac{1}{81} = \dfrac{1}{3^4} \)

Answer 8

Okay.

Answer 9

The question now is:

\(3^n = \dfrac{1}{3^4} \)

Now we use the definition of a negative exponent:

\(a^{-n} = \dfrac{1}{a^n} \)

Answer 10

We start from the right side of the definition:

\(\dfrac{1}{a^n} = a^{-n} \)

If we use the fraction from your problem, and we follow the definition, we get:

\(\dfrac{1}{3^4} = 3^{-4} \)

Answer 11

The exponent, n, is -4.
Answer: C. -4

Answer 12

I get it now. Thanks!