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x^((3)/(2))*((1)/(x^((5)/(2))))^(2)
Which of the following is equivalent to the given expression for all positive values of
x ?
Choose 1 answer:
(A)
(1)/(x^(2))
(B)
(1)/(sqrt(x^(7)))
(C)
(1)/(root(7)(x^(2)))
(D)
(1)/(root(4)(x^(19)))

$x^{\frac{3}{2}} \cdot\left(\frac{1}{x^{\frac{5}{2}}}\right)^{2}$ $\newline$ Which of the following is equivalent to the given expression for all positive values of $x$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{x^{2}}$ $\newline$ (B) $\frac{1}{\sqrt{x^{7}}}$ $\newline$ (C) $\frac{1}{\sqrt[7]{x^{2}}}$ $\newline$ (D) $\frac{1}{\sqrt[4]{x^{19}}}$

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Compare linear and exponential growth

Full solution

Q.

x^{\frac{3}{2}} \cdot\left(\frac{1}{x^{\frac{5}{2}}}\right)^{2}

\newline

Which of the following is equivalent to the given expression for all positive values of

x

?

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

\frac{1}{\sqrt{x^{7}}}

\newline

(C)

\frac{1}{\sqrt[7]{x^{2}}}

\newline

(D)

\frac{1}{\sqrt[4]{x^{19}}}

Simplify expression inside parentheses: Simplify the given expression using the properties of exponents. $\newline$ The given expression is $x^{\frac{3}{2}} \cdot \left(\frac{1}{x^{\frac{5}{2}}}\right)^2$ . We can simplify the expression inside the parentheses first by raising $x^{\frac{5}{2}}$ to the power of $2$ , which means we multiply the exponents. $\newline$ $\left(\frac{1}{x^{\frac{5}{2}}}\right)^2 = \frac{1}{x^{\frac{5}{2} \cdot 2}} = \frac{1}{x^5}.$
Multiply $x^{\frac{3}{2}}$ by $\frac{1}{x^5}$ : Now, we multiply $x^{\frac{3}{2}}$ by $\frac{1}{x^5}$ . $\newline$ Using the property of exponents that states when we divide like bases we subtract the exponents, we get: $\newline$ $x^{\frac{3}{2}} \cdot \frac{1}{x^5} = x^{\frac{3}{2} - 5} = x^{-\frac{7}{2}}$ .
Convert negative exponent to positive: Convert the negative exponent to a positive exponent by taking the reciprocal. $\newline$ $x^{(-\frac{7}{2})}$ is equivalent to $\frac{1}{x^{(\frac{7}{2})}}$ .
Convert fractional exponent to radical: Simplify the expression further by converting the fractional exponent to a radical. $\newline$ $x^{\frac{7}{2}}$ is equivalent to the square root of $x^7$ , which can also be written as $\sqrt{x^7}$ . $\newline$ Therefore, $\frac{1}{x^{\frac{7}{2}}}$ is equivalent to $\frac{1}{\sqrt{x^7}}$ .
Match with answer choices: Match the simplified expression to the given answer choices. $\newline$ The expression $\frac{1}{\sqrt{x^7}}$ matches with choice (B) $\frac{1}{\sqrt{x^7}}$ .

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