If x is a real number greater than 1 satisfying the exponential equation \left(x^x\right)^… — GRE General Test

x

is a real number greater than

1

satisfying the exponential equation

\left(x^x\right)^{\sqrt{x}} = x^{\left(x^{\sqrt{x}}\right)}

what is the value of

x

The value of

x

\frac{9}{4}

Applying the exponent rule

(a^b)^c = a^{bc}

simplifies the left side to

x^{x\sqrt{x}}

. Since the base

x > 1

is identical on both sides, we set the exponents equal:

x\sqrt{x} = x^{\sqrt{x}}

. Expressing

x\sqrt{x}

x^{3/2}

yields

x^{3/2} = x^{\sqrt{x}}

, which implies

\sqrt{x} = \frac{3}{2}

. Squaring both sides gives

x = \frac{9}{4}

Apply the power rule

(a^b)^c = a^{bc}

to the left-hand side of the equation.

\left(x^x\right)^{\sqrt{x}} = x^{x \cdot \sqrt{x}} = x^{x^{3/2}}

Raising a power to another exponent requires multiplying the exponents:

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

Equate the exponents of the expressions on both sides, as the bases are equal and

x > 1

x \sqrt{x} = x^{\sqrt{x}}

, which means

x^{3/2} = x^{\sqrt{x}}

x^A = x^B

and

x > 1

, then

A = B

Equate exponents once more for the base

x

\frac{3}{2} = \sqrt{x}

Since the bases are identical (

x > 1

), their exponents must be equal.

Square both sides to solve for

x

x = \left(\frac{3}{2}\right)^2 = \frac{9}{4}

Squaring

\sqrt{x}

isolates

x

Properties of exponents and nested power rules with radicals

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