If x is a real number such that \sqrt[4]{3^x + 3^x + 3^x} = 27, what is the value of x? — GMAT

If $x$ is a real number such that $\sqrt[4]{3^x + 3^x + 3^x} = 27$ , what is the value of $x$ ?

Answer: 11

The value of

x

is 11.

Summing three terms of

3^x

yields

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

. Expressing the fourth root of

3^{x+1}

as a fractional power gives

3^{\frac{x+1}{4}}

. Equating this to

27 = 3^3

results in

\frac{x+1}{4} = 3

, leading directly to

x = 11

Combine the repeated addition terms inside the fourth root.

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

Adding three identical terms

3^x

is equivalent to multiplying

3^x

by 3, which increments the exponent by 1.

Rewrite the fourth root expression using a fractional exponent.

\sqrt[4]{3^{x+1}} = 3^{\frac{x+1}{4}}

Applying the property

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

Express the constant on the right side of the equation with base 3.

27 = 3^3

Rewriting both sides of the equation with the same base allows equating exponents.

Equate the exponents and solve for

x

\frac{x+1}{4} = 3 \implies x + 1 = 12 \implies x = 11

Since the bases are equal, the powers must be equal.

Combining like exponential terms and equating fractional exponents

Estimated Time:1m 30s

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