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

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

Answer: 2

The value of

x

2

By rewriting

27

3^3

and

9

3^2

, the equation becomes

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

. Applying the power rule of exponents, this simplifies to

3^{3x} = 3^{2x+2}

. Since the bases are now the same, the exponents can be set equal to each other:

3x = 2x + 2

. Solving for

x

gives

x = 2

Rewrite

27

and

9

as powers of

3

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

To solve exponential equations with different bases, it is helpful to express them using a common base.

Apply the power of a power rule

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

to simplify the exponents.

3^{3x} = 3^{2x + 2}

This simplifies the exponential expressions on both sides of the equation.

Set the exponents equal to each other.

3x = 2x + 2

Since the bases are equal (

3 = 3

), the exponents must also be equal for the equation to hold.

Solve the linear equation for

x

x = 2

Subtracting

2x

from both sides isolates the variable

x

Solving exponential equations by expressing both sides with a common base and equating exponents.

Estimated Time:45s

Question