For what value of x is the equation 8^{x + 1} = 16^x true? — SAT

For what value of $x$ is the equation $8^{x + 1} = 16^x$ true?

Answer: 3

The correct answer is 3.

The correct answer is 3. To find the value of

x

, express both 8 and 16 as powers of 2:

8 = 2^3

and

16 = 2^4

. Substituting these values into the equation gives

(2^3)^{x + 1} = (2^4)^x

. Applying the power of a power rule,

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

, yields

2^{3(x + 1)} = 2^{4x}

, which simplifies to

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

. Since the bases are now the same, their exponents must be equal:

3x + 3 = 4x

. Subtracting

3x

from both sides gives

x = 3

Express the bases 8 and 16 as powers of 2.

(2^3)^{x + 1} = (2^4)^x

Writing both sides of the equation with a common base allows for the equating of exponents.

Apply the power of a power rule

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

to simplify the exponents.

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

This simplifies each side of the equation to a single base with a single exponent.

Equate the exponents and solve the resulting linear equation.

3x + 3 = 4x

, which simplifies to

x = 3

Since the bases are equal, their exponents must be equal for the equation to hold true.

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

An alternative approach is to write the equation in terms of base 4. While 8 is not an integer power of 4, we can write

8 = 4^{1.5}

and

16 = 4^2

. The equation becomes

(4^{1.5})^{x+1} = (4^2)^x

, which simplifies to

1.5(x+1) = 2x

. Solving this gives

1.5x + 1.5 = 2x \implies 0.5x = 1.5 \implies x = 3

Estimated Time:45s

Question