If 8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}, what is the value of x? — SAT

If $8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}$ , what is the value of $x$ ?

\frac{8}{3}

To solve the equation

8^{x+2} = \left(\frac{1}{4}\right)^{1-3x}

, both bases can be written as powers of

2

because

8 = 2^3

and

\frac{1}{4} = 2^{-2}

. Substituting these values into the equation yields

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

. Applying the power rule of exponents,

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

, we multiply the exponents to get

2^{3x+6} = 2^{-2+6x}

. Since the bases are equal, we can set the exponents equal to each other:

3x + 6 = -2 + 6x

. Solving this linear equation by subtracting

3x

from both sides and adding

2

to both sides gives

8 = 3x

, which simplifies to

x = \frac{8}{3}

. Thus, the option with the value

\frac{8}{3}

is correct.

Express both bases in the equation,

8

and

\frac{1}{4}

, as powers of

2

The base

8

is written as

2^3

and the base

\frac{1}{4}

is written as

2^{-2}

, yielding the equation

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

Expressing exponential terms with a common base is necessary to equate and solve their exponents.

Apply the power of a power exponent rule,

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

, to simplify both sides of the equation.

The equation becomes

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

, which simplifies to

2^{3x+6} = 2^{-2+6x}

This simplifies each side to a single base with a single exponent.

Since the bases are equal, set the exponents equal to each other and solve the resulting linear equation for

x

3x + 6 = -2 + 6x \implies 8 = 3x \implies x = \frac{8}{3}

Two exponential expressions with the same positive base are equal if and only if their exponents are equal.

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Question