If 9^{2x + 1} = 27^{x - 3}, what is the value of x? — SAT

If $9^{2x + 1} = 27^{x - 3}$ , what is the value of $x$ ?

-11

To solve

9^{2x + 1} = 27^{x - 3}

, express 9 and 27 with a common base of 3:

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

. Apply the power of a power rule to get

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

, which simplifies to

3^{4x + 2} = 3^{3x - 9}

. Equating the exponents gives

4x + 2 = 3x - 9

. Solving for

x

by subtracting

3x

and

2

from both sides yields

x = -11

. Therefore, the value of

x

-11

Rewrite both sides of the equation with a common base.

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

Since 9 and 27 are both powers of 3 (

9 = 3^2

and

27 = 3^3

), expressing them with the same base allows the application of exponent rules.

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

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

, which simplifies to

3^{4x+2} = 3^{3x-9}

Multiplying the inner exponent by the outer exponent simplifies the expression on both sides.

Set the exponents equal to each other.

4x + 2 = 3x - 9

Since the bases are equal and positive, the exponential expressions are equal if and only if their exponents are equal.

Solve the linear equation for

x

x = -11

Subtracting

3x

and

2

from both sides isolates the variable

x

Solving exponential equations by converting to a common base and equating exponents.

Question