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sqrt(8^x / 2^x) = 4, find x

June 29, 2026Mathematics

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Answer

2

7 steps

Start with the given equation.
$\sqrt{\frac{8^x}{2^x}} = 4$
Square both sides of the equation to eliminate the square root.
$\left(\sqrt{\frac{8^x}{2^x}}\right)^2 = 4^2$ $\frac{8^x}{2^x} = 16$
Rewrite the bases 8 and 16 as powers of 2.
$\frac{(2^3)^x}{2^x} = 2^4$
Apply the exponent rule $(a^m)^n = a^{mn}$ to the numerator.
$\frac{2^{3x}}{2^x} = 2^4$
Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ to the left side.
$2^{3x-x} = 2^4$ $2^{2x} = 2^4$
Since the bases are equal, equate the exponents.
$2x = 4$
Solve for $x$ by dividing both sides by 2.
$x = \frac{4}{2}$ $x = 2$

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