Simplify: $$ \left[\frac{3^{x+1}+12.3^{x-1}}{7.9^{x}}\right]^{\frac{1}{x}} $$

Question

UpStudy AI · Accepted Answer

We start with the expression

[ (3^(x+1) + 12·3^(x–1)) / (7·9^x) ]^(1/x).

Note: The dot “·” represents multiplication. That is, 12·3^(x–1) means 12 times 3^(x–1) and 7·9^x means 7 times 9^x.

Step 1. Rewrite Numerator

Write 3^(x+1) as 3^(x–1)·3^2 (since 3^(x+1) = 3^(x–1)·3^2). Then the numerator becomes:

3^(x–1)·3^2 + 12·3^(x–1)
  = 3^(x–1)·(9 + 12)
  = 3^(x–1)·21.

Step 2. Rewrite Denominator

Express 9^x as (3^2)^x = 3^(2x). So the denominator becomes:

7·9^x = 7·3^(2x).

Step 3. Combine and simplify the fraction

Our expression inside the bracket now is:

[21·3^(x–1)] / [7·3^(2x)].

Divide the constant coefficients:

21 / 7 = 3.

Now, handle the powers of 3 by subtracting exponents:

3^(x–1) / 3^(2x) = 3^[(x–1) – 2x] = 3^(–x – 1).

So the simplified fraction is:

3 · 3^(–x – 1).

Combine the powers (recall 3 = 3^1):

3^1 · 3^(–x – 1) = 3^(1 – x – 1) = 3^(–x).

Thus, the expression inside the bracket becomes 3^(–x).

Step 4. Apply the exponent 1/x

Now take the 1/x-th power of the expression:

[3^(–x)]^(1/x) = 3^(–x/x) = 3^(–1).

Step 5. Write the Final Answer

3^(–1) equals 1/3.

Therefore, the simplified expression is 1/3.
The simplified expression is $$ \frac{1}{3} $$.

Simplify: \( \left[\frac{3^{x+1}+12.3^{x-1}}{7.9^{x}}\right]^{\frac{1}{x}} \)