Question

what is the solution to the equation 9^(x+1) =27

Answer 1

ANSWER

$x = \frac{1}{2}$

EXPLANATION

The given exponential equation is

${9}^{x + 1} = 27$

The greatest common factor of 9 and 27 is 3.

We rewrite the each side of the equation to base 3.

${3}^{2(x + 1)} = {3}^{3}$

Since the bases are equal, we can equate the exponents.

$2(x + 1) = 3$

Expand the parenthesis to get:

$2x + 2 = 3$

Group similar terms

$2x = 3 - 2$

$2x = 1$

$x = \frac{1}{2}$

Answer 2

For this case we must solve the following equation:

$9 ^ {x + 1} = 27$

We rewrite:

$9 = 3 * 3 = 3 ^ 2\\27 = 3 * 3 * 3 = 3 ^ 3$

Then the expression is:

$3^ {2 (x + 1)} = 3 ^ 3$

Since the bases are the same, the two expressions are only equal if the exponents are also equal. So, we have:

$2 (x + 1) = 3$

We apply distributive property to the terms within parentheses:

$2x + 2 = 3$

Subtracting 2 on both sides of the equation:

$2x = 3-2\\2x = 1$

Dividing between 2 on both sides of the equation:

$x = \frac {1} {2}$

Answer:

$x = \frac {1} {2}$