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sqrtx*sqrt((y^(5))/(x^(3)))
Which of the following is equivalent to the given expression for all positive values of
x and
y ?
Choose 1 answer:
(A)
x^(-2)y^(5)
(B)
x^(-1)y^((5)/(2))
(C)
xy^((5)/(2))
(D)
x^(2)y^(5)

$\sqrt{x} \cdot \sqrt{\frac{y^{5}}{x^{3}}}$ $\newline$ Which of the following is equivalent to the given expression for all positive values of $x$ and $y$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $x^{-2} y^{5}$ $\newline$ (B) $x^{-1} y^{\frac{5}{2}}$ $\newline$ (C) $x y^{\frac{5}{2}}$ $\newline$ (D) $x^{2} y^{5}$

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Compare linear and exponential growth

Full solution

Q.

\sqrt{x} \cdot \sqrt{\frac{y^{5}}{x^{3}}}

\newline

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

x

and

y

?

\newline

Choose

1

answer:

\newline

(A)

x^{-2} y^{5}

\newline

(B)

x^{-1} y^{\frac{5}{2}}

\newline

(C)

x y^{\frac{5}{2}}

\newline

(D)

x^{2} y^{5}

Express in Exponents: We start by expressing the given expression in terms of exponents to simplify it. The square root of a number is the same as raising that number to the power of $1/2$ . So, we rewrite $\sqrt{x}$ as $x^{1/2}$ and $\sqrt{\left(\frac{y^{5}}{x^{3}}\right)}$ as $\left(\frac{y^{5}}{x^{3}}\right)^{1/2}$ .
Apply Exponent Rule: Next, we apply the exponent rule $(a^{(m/n)})^{(p/q)} = a^{((m*p)/(n*q))}$ to the second part of the expression. This means $((y^{5})/(x^{3}))^{(1/2)}$ becomes $y^{((5*1)/(2*1))}/x^{((3*1)/(2*1))}$ , which simplifies to $y^{(5/2)}/x^{(3/2)}$ .
Multiply Expressions: Now, we multiply the two parts together: $x^{\frac{1}{2}} \times \left(\frac{y^{\frac{5}{2}}}{x^{\frac{3}{2}}}\right)$ . When multiplying expressions with the same base, we add the exponents, according to the rule $a^m \times a^n = a^{m+n}$ .
Simplify Exponents: Applying the rule of adding exponents, we get $x^{\frac{1}{2} - \frac{3}{2}} * y^{\frac{5}{2}}$ . Simplifying the exponents for $x$ gives us $x^{-1} * y^{\frac{5}{2}}$ .
Final Simplified Expression: The simplified expression is $x^{-1}y^{\frac{5}{2}}$ , which matches option (B) $x^{-1}y^{\left(\frac{5}{2}\right)}$ .

More problems from Compare linear and exponential growth

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\newline

\newline

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\newline

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Question

f(x)

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\newline

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\newline

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h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

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Question

p(x)

q(x)

are differentiable.

\newline

r(x)

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\newline

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\newline

Simplify any fractions.

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Question

u(x)

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\newline

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\newline

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\newline

Simplify any fractions.

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Question

a(x)

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\newline

c(x)

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\newline

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c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

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m(x)

n(x)

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\newline

o(x)

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\newline

m(7)= 2

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o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

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