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(sqrt(8c^(20)))/(sqrt(32c^(4)))
Which of the following is equivalent to the given expression for all
c!=0 ?
Choose 1 answer:
(A)
(1)/(2)c^(4)
(B)
(1)/(2)c^(8)
(C)
(1)/(4)c^(16)
D
(1)/(4)c^(2)sqrtc

$\frac{\sqrt{8 c^{20}}}{\sqrt{32 c^{4}}}$ $\newline$ Which of the following is equivalent to the given expression for all $c \neq 0$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2} c^{4}$ $\newline$ (B) $\frac{1}{2} c^{8}$ $\newline$ (C) $\frac{1}{4} c^{16}$ $\newline$ D $\frac{1}{4} c^{2} \sqrt{c}$

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Gaussian integral

Full solution

Q.

\frac{\sqrt{8 c^{20}}}{\sqrt{32 c^{4}}}

\newline

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

c \neq 0

?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2} c^{4}

\newline

(B)

\frac{1}{2} c^{8}

\newline

(C)

\frac{1}{4} c^{16}

\newline

D

\frac{1}{4} c^{2} \sqrt{c}

Simplify Square Roots: Simplify the square roots separately. $\sqrt{8c^{20}}$ can be simplified by taking the square root of $8$ and $c^{20}$ separately. The square root of $8$ is $2\sqrt{2}$ and the square root of $c^{20}$ is $c^{10}$ because $(c^{20})^{1/2} = c^{20/2} = c^{10}$ . Similarly, $\sqrt{32c^{4}}$ can be simplified by taking the square root of $32$ and $8$ $0$ separately. The square root of $32$ is $8$ $2$ and the square root of $8$ $0$ is $8$ $4$ because $8$ $5$ .
Divide Simplified Square Roots: Divide the simplified square roots. $\newline$ Now we have $(2\sqrt{2}c^{10}) / (4\sqrt{2}c^{2})$ . $\newline$ We can cancel out the common terms $2\sqrt{2}$ from the numerator and the denominator. $\newline$ This leaves us with $c^{10} / c^{2}$ .
Apply Quotient Rule: Apply the quotient rule for exponents. $c^{10} / c^{2}$ can be simplified by subtracting the exponents because $c^{a} / c^{b} = c^{a-b}$ . So, $c^{10} / c^{2} = c^{10-2} = c^{8}$ .
Write Final Expression: Write the final simplified expression. $\newline$ The final simplified expression is $c^{8}$ . $\newline$ Now we need to match this with the given answer choices.
Match with Answer Choices: Match the simplified expression with the answer choices. $\newline$ The expression $c^{8}$ matches with answer choice (B) $\frac{1}{2}c^{8}$ . $\newline$ However, we need to ensure that the coefficient $\frac{1}{2}$ is correct.
Check Coefficient: Check the coefficient. $\newline$ We initially simplified $\sqrt{8}$ to $2\sqrt{2}$ and $\sqrt{32}$ to $4\sqrt{2}$ , and these terms canceled each other out, leaving no coefficient. $\newline$ Therefore, the coefficient should be $1$ , not $\frac{1}{2}$ . $\newline$ This means there is a mistake in our previous steps.

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