If 27^{x-1} = \sqrt{3^{5x+1}}, what is the value of x? — SAT

If $27^{x-1} = \sqrt{3^{5x+1}}$ , what is the value of $x$ ?

Answer: 7

The correct answer is 7.

The correct answer is 7. By converting the base of 27 to

3^3

and rewriting the radical on the right side as a fractional exponent of

\frac{1}{2}

, the equation becomes

3^{3(x-1)} = 3^{\frac{5x+1}{2}}

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

3x - 3 = \frac{5x+1}{2}

. Multiplying both sides by 2 yields

6x - 6 = 5x + 1

. Subtracting

5x

and adding 6 to both sides isolates

x

, giving the solution

x = 7

Express both sides of the equation with a common base of 3.

(3^3)^{x-1} = (3^{5x+1})^{\frac{1}{2}}

Expressing terms with a common base allows the exponents to be compared directly.

Apply the power of a power exponent rule,

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

, to simplify the exponents on both sides.

3^{3x-3} = 3^{\frac{5x+1}{2}}

Simplifying the expressions makes it possible to set the exponent expressions equal to each other.

Equate the exponents since the bases are identical.

3x - 3 = \frac{5x+1}{2}

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must also be equal.

Solve the linear equation for

x

by clearing the fraction and isolating the variable.

x = 7

Multiplying both sides by 2 gives

6x - 6 = 5x + 1

. Subtracting

5x

from both sides and adding 6 to both sides isolates

x

, resulting in

x = 7

Solving exponential equations by converting to a common base and applying exponent laws.

Question