If x > 0, which of the following is equivalent to the expression \sqrt{9x^4 + 16x^4}? — GRE General Test

If $x > 0$ , which of the following is equivalent to the expression $\sqrt{9x^4 + 16x^4}$ ?

The correct answer is

5x^2

First combine the terms under the square root to get

\sqrt{25x^4}

. Taking the square root yields

\sqrt{25} \cdot \sqrt{x^4} = 5x^2

. Thus,

5x^2

is the correct simplified expression.

Combine like terms inside the square root.

9x^4 + 16x^4 = 25x^4

Since both terms have the same variable factor

x^4

, their coefficients can be added directly.

Apply the product rule for radicals:

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

\sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4}

The square root of a product is equal to the product of the square roots.

Evaluate the square root of the coefficient and the variable term.

5 \cdot x^2 = 5x^2

\sqrt{25} = 5

and

\sqrt{x^4} = (x^4)^{1/2} = x^2

(since

x > 0$).

Simplifying algebraic radicals by combining like terms under the square root and applying power rules for exponents.

Soru