For all positive real numbers a and b such that b = 4a, if a^b = b^a, then the value of a … — GMAT

For all positive real numbers $a$ and $b$ such that $b = 4a$ , if $a^b = b^a$ , then the value of $a$ is $\sqrt{2}$ .

Answer: Answer

False. The statement is false because the exact value of

a

\sqrt[3]{4}

, which is not equal to

\sqrt{2}

The statement is false because simplifying

a^{4a} = (4a)^a

leads directly to

a^3 = 4

, giving

a = \sqrt[3]{4} = 2^{2/3}

. The claimed value

\sqrt{2} = 2^{1/2}

fails to satisfy the original equation, making the assertion mathematically false.

Substitute

b = 4a

into the given exponent equation

a^b = b^a

a^{4a} = (4a)^a

Eliminate variable

b

to express the equation solely in terms of

a

Raise both sides of the equation to the power of

\frac{1}{a}

(a^{4a})^{1/a} = ((4a)^a)^{1/a} \implies a^4 = 4a

Apply power of a power exponent rule

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

to simplify the exponents.

Divide both sides by

a

(since

a > 0

) and solve for

a

\frac{a^4}{a} = \frac{4a}{a} \implies a^3 = 4 \implies a = \sqrt[3]{4} = 2^{2/3}

Isolate

a

using standard division and radical extraction rules.

Compare the calculated value of

a

with

\sqrt{2}

a = 2^{2/3} \approx 1.587

, whereas

\sqrt{2} = 2^{1/2} \approx 1.414

. Thus

a \neq \sqrt{2}

Determine the truth value of the claimed conclusion.

Solving variable exponent equations of the form

a^b = b^a

using exponent power rules and root extraction.

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