For x > 0, which of the following is equivalent to the expression \frac{\sqrt{x^3 \cdot \s… — SAT

For $x > 0$ , which of the following is equivalent to the expression $\frac{\sqrt{x^3 \cdot \sqrt{x}}}{\sqrt[4]{x^3}}$ ?

A
$x^4$
B
$x\sqrt[4]{x}$
$x$ Answer
D
$x^2\sqrt{x}$

Answer

The expression is equivalent to

x

To simplify the expression, convert each radical into an expression with a rational exponent. The inner term of the numerator contains

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

. Multiplying this by

x^3

yields

x^3 \cdot x^{1/2} = x^{3 + 1/2} = x^{7/2}

. Applying the outer square root to this product yields

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

. The denominator is

\sqrt[4]{x^3} = x^{3/4}

. Dividing the simplified numerator by the denominator is performed by subtracting the exponent of the denominator from that of the numerator:

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

. Therefore, the expression simplifies to

x

Step-by-Step Solution

Convert the inner radical in the numerator into a fractional exponent.

The term

\sqrt{x}

becomes

x^{1/2}

, so the product inside the outer square root becomes

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

Converting radicals to rational exponents allows the application of standard exponent properties.

Combine the terms inside the outer square root using exponent addition rules.

Since

3 + 1/2 = 7/2

, the product is

x^{7/2}

When multiplying exponential terms with the same base, their exponents are added.

Simplify the entire numerator by applying the outer square root.

The numerator becomes

\sqrt{x^{7/2}} = (x^{7/2})^{1/2} = x^{7/4}

Applying a square root to a term is equivalent to raising that term to the power of

1/2

Convert the denominator into a fractional exponent.

The denominator

\sqrt[4]{x^3}

becomes

x^{3/4}

Converting the denominator's radical to a rational exponent allows division with the numerator.

Divide the numerator by the denominator using exponent subtraction rules.

The final expression is

x^{7/4} / x^{3/4} = x^{7/4 - 3/4} = x^{4/4} = x

When dividing exponential terms with the same base, the exponent of the denominator is subtracted from the exponent of the numerator.

Key Concept

Simplification of equivalent algebraic expressions involving radical and rational exponents.

Alternative Method

An alternative approach is to substitute a convenient value for

x

that is a perfect power of 4, such as

x = 16

. The numerator becomes

\sqrt{16^3 \cdot \sqrt{16}} = \sqrt{4096 \cdot 4} = \sqrt{16384} = 128

. The denominator becomes

\sqrt[4]{16^3} = \sqrt[4]{4096} = 8

. Evaluating the fraction yields

128 / 8 = 16

. Since substituting

x = 16

yields a value of

16

, the expression must be equivalent to

x

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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