An expression is given as \sqrt{2^6 \cdot 3^4}. What is the value of this expression? — ACT

An expression is given as $\sqrt{2^6 \cdot 3^4}$ . What is the value of this expression?

The correct answer of

72

is obtained by applying the rules of exponents under a radical. Evaluating

\sqrt{2^6 \cdot 3^4}

simplifies to

2^3 \cdot 3^2

, which equals

8 \cdot 9 = 72

Express the square root as a fractional exponent of 1/2.

(2^6 \cdot 3^4)^{\frac{1}{2}}

The square root of any non-negative number

x

can be written as

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

Apply the power of a product rule

(ab)^n = a^n b^n

and power of a power rule

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

to simplify the exponents.

2^{6 \cdot \frac{1}{2}} \cdot 3^{4 \cdot \frac{1}{2}} = 2^3 \cdot 3^2

Exponents are multiplied when raising a power to a power.

Evaluate the simplified exponential expressions and multiply the results.

8 \cdot 9 = 72

Evaluating

2^3 = 8

and

3^2 = 9

and calculating their product gives the final simplified value.

Simplifying expressions containing exponents and square roots

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