For all positive values of x, which of the following is equivalent to the expression 3x^{\… — SAT

For all positive values of $x$ , which of the following is equivalent to the expression $3x^{\frac{1}{2}} (2x^{\frac{3}{2}} - x^{-\frac{1}{2}})$ ?

6x^2 - 3

Distributing the term

3x^{\frac{1}{2}}

to both terms inside the parentheses yields

3x^{\frac{1}{2}}(2x^{\frac{3}{2}}) - 3x^{\frac{1}{2}}(x^{-\frac{1}{2}})

. Multiplying the coefficients and adding the exponents according to the rule

x^a \cdot x^b = x^{a+b}

gives

6x^{\frac{1}{2} + \frac{3}{2}} - 3x^{\frac{1}{2} - \frac{1}{2}} = 6x^2 - 3x^0

. Since

x^0 = 1

for any positive

x

, the simplified equivalent expression is

6x^2 - 3

Distribute the term

3x^{\frac{1}{2}}

to both terms inside the parentheses.

3x^{\frac{1}{2}} \cdot 2x^{\frac{3}{2}} - 3x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}}

Apply the distributive property

a(b - c) = ab - ac

to expand the expression.

Multiply the coefficients and apply the product rule for exponents,

x^a \cdot x^b = x^{a+b}

, to each product.

(3 \cdot 2)x^{\frac{1}{2} + \frac{3}{2}} - 3x^{\frac{1}{2} + (-\frac{1}{2})}

When multiplying terms with the same base, keep the base and add the exponents.

Simplify the arithmetic in the exponents and evaluate the resulting terms.

6x^2 - 3

Since

\frac{1}{2} + \frac{3}{2} = 2

and

\frac{1}{2} - \frac{1}{2} = 0

, the expression simplifies to

6x^2 - 3x^0

. Because

x > 0

x^0 = 1

, making the final expression

6x^2 - 3

Equivalent Algebraic Expressions

Substitute a simple value for

x

, such as

x = 4

. The original expression evaluates to

3(4)^{1/2}(2(4)^{3/2} - 4^{-\frac{1}{2}}) = 3(2)(2(8) - \frac{1}{2}) = 6(16 - 0.5) = 6(15.5) = 93

. Evaluating the correct expression

6x^2 - 3

x = 4

yields

6(16) - 3 = 96 - 3 = 93

. Evaluating the other options at

x = 4

yields different values.

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