For all positive real numbers x, the nested radical expression \sqrt{x\sqrt{x}} is mathema… — GMAT

For all positive real numbers $x$ , the nested radical expression $\sqrt{x\sqrt{x}}$ is mathematically equivalent to $x^{\frac{3}{4}}$ .

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The statement is true because expressing

\sqrt{x\sqrt{x}}

in fractional exponent form yields

x^{\frac{3}{4}}

for all positive real numbers

x

The radical expression

\sqrt{x\sqrt{x}}

simplifies to

x^{\frac{3}{4}}

by systematically applying exponent laws: the inner root gives

x^{\frac{1}{2}}

, multiplying by

x

gives

x^{\frac{3}{2}}

, and applying the outer root raises

x^{\frac{3}{2}}

to the

\frac{1}{2}

power, producing

x^{\frac{3}{4}}

Express the inner radical as a fractional exponent.

\sqrt{x} = x^{\frac{1}{2}}

, making the expression under the outer root

x \cdot x^{\frac{1}{2}}

By definition of fractional exponents,

\sqrt[n]{a^m} = a^{\frac{m}{n}}

Combine the terms inside the outer radical using the product rule for exponents.

x^1 \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}

When multiplying terms with the same base, add their exponents.

Apply the outer square root as an exponent of

\frac{1}{2}

and simplify using the power rule.

\sqrt{x^{\frac{3}{2}}} = (x^{\frac{3}{2}})^{\frac{1}{2}} = x^{\frac{3}{2} \cdot \frac{1}{2}} = x^{\frac{3}{4}}

Raising a power to another power requires multiplying the exponents.

Simplification of nested radical expressions using fractional exponent laws

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