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sqrt(y^(3)z)*sqrt(y^(13)z^(8))
Which of the following is equivalent to the given expression for all real values of
y and
z ?
Choose 1 answer:
(A)
y^(4)z^(3)
(B)
y^(8)z^(4)
(C)
y^(8)z^(4)sqrtz
(D)
y^(15)z^(4)sqrty

$\sqrt{y^{3} z} \cdot \sqrt{y^{13} z^{8}}$ $\newline$ Which of the following is equivalent to the given expression for all real values of $y$ and $z$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $y^{4} z^{3}$ $\newline$ (B) $y^{8} z^{4}$ $\newline$ (C) $y^{8} z^{4} \sqrt{z}$ $\newline$ (D) $y^{15} z^{4} \sqrt{y}$

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Compare linear and exponential growth

Full solution

Q.

\sqrt{y^{3} z} \cdot \sqrt{y^{13} z^{8}}

\newline

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

y

and

z

?

\newline

Choose

1

answer:

\newline

(A)

y^{4} z^{3}

\newline

(B)

y^{8} z^{4}

\newline

(C)

y^{8} z^{4} \sqrt{z}

\newline

(D)

y^{15} z^{4} \sqrt{y}

Property of square roots: We are given the expression $\sqrt{y^{3}z} \cdot \sqrt{y^{13}z^{8}}$ . To simplify this expression, we will use the property of square roots that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ .
Multiplying the square roots: Now we multiply the two square roots together: $\sqrt{y^{3}z} \times \sqrt{y^{13}z^{8}} = \sqrt{(y^{3}z) \times (y^{13}z^{8})}$ .
Combining terms under the square root: Next, we combine the terms under the square root: $\sqrt{(y^{3}z) \cdot (y^{13}z^{8})} = \sqrt{y^{3+13} \cdot z^{1+8}}$ .
Adding exponents of like bases: We add the exponents of like bases: $\sqrt{y^{(3+13)} \cdot z^{(1+8)}} = \sqrt{y^{16} \cdot z^{9}}$ .
Simplifying the square root of each term: Now we simplify the square root of each term: $\sqrt{y^{16} \cdot z^{9}} = \sqrt{y^{16}} \cdot \sqrt{z^{9}}$ .
Dividing exponents by $2$ : Since $y^{16}$ and $z^{9}$ are perfect squares, we can take the square root of each: $\sqrt{y^{16}} \cdot \sqrt{z^{9}} = y^{\frac{16}{2}} \cdot z^{\frac{9}{2}}$ .
Recognizing $z^{4.5}$ as $z^{4}\sqrt{z}$ : We simplify the exponents by dividing them by $2$ : $y^{\frac{16}{2}} \cdot z^{\frac{9}{2}} = y^{8} \cdot z^{4.5}$ .
Matching with option (C): We recognize that $z^{4.5}$ is the same as $z^{4}\sqrt{z}$ : $y^{8} \cdot z^{4.5} = y^{8} \cdot z^{4} \cdot \sqrt{z}$ .
Matching with option (C): We recognize that $z^{4.5}$ is the same as $z^{4}\sqrt{z}$ : $y^{8} \cdot z^{4.5} = y^{8} \cdot z^{4} \cdot \sqrt{z}$ . We compare our result with the given answer choices and find that it matches with option (C) $y^{8}z^{4}\sqrt{z}$ .

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\newline

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\newline

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\newline

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\newline

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