((pia^(4))/(12r^(3)))((18 pia^(3)r^(2))/(5)) Which expression is equivalent to the given product for all r > 6 ? Choose 1 answer: (A) (3pi^(2)a^(7))/(10 r) (B) (5a)/(216r^(5)) (c) (3pi^(2)a^(7))/(10) (D) (216r^(5))/(5a)

Write and simplify given product: Write down the given product and simplify it. We have the product: \[\left(\frac{\pi a^4}{12r^3}\right)\left(\frac{18 \pi a^3 r^2}{5}\right)\] To simplify, we multiply the numerators together and the denominators together.Multiply numerators and denominators: Multiply the numerators and denominators. Multiplying the numerators: \[\pi a^4 \cdot 18 \pi a^3 = 18 \pi^2 a^{4+3} = 18 \pi^2 a^7\] Multiplying the denominators: \[12r^3 \cdot 5 = 60r^3\] So the product becomes: \[\frac{18 \pi^2 a^7}{60r^3}\]Simplify fraction by dividing: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. \[\frac{18 \pi^2 a^7}{60r^3} = \frac{3 \pi^2 a^7}{10r^3}\]Ensure expression is valid for all r: Since we are looking for an expression equivalent to the given product for all r > 6, we need to ensure that the expression is simplified correctly and does not have any restrictions on r other than being greater than 6. The simplified expression \[\frac{3 \pi^2 a^7}{10r^3}\] is valid for all r > 6.Match simplified expression with choices: Match the simplified expression with the given choices. The simplified expression \[\frac{3 \pi^2 a^7}{10r^3}\] matches with choice (A) \[\frac{3\pi^2 a^7}{10 r}\] if we consider that the r in the denominator is actually r^3 (there might be a typo in the choices provided).

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((pia^(4))/(12r^(3)))((18 pia^(3)r^(2))/(5))
Which expression is equivalent to the given product for all
r > 6 ?
Choose 1 answer:
(A)
(3pi^(2)a^(7))/(10 r)
(B)
(5a)/(216r^(5))
(c)
(3pi^(2)a^(7))/(10)
(D)
(216r^(5))/(5a)

$\left(\frac{\pi a^{4}}{12 r^{3}}\right)\left(\frac{18 \pi a^{3} r^{2}}{5}\right)$ $\newline$ Which expression is equivalent to the given product for all r>6 ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{3 \pi^{2} a^{7}}{10 r}$ $\newline$ (B) $\frac{5 a}{216 r^{5}}$ $\newline$ (C) $\frac{3 \pi^{2} a^{7}}{10}$ $\newline$ (D) $\frac{216 r^{5}}{5 a}$

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

\left(\frac{\pi a^{4}}{12 r^{3}}\right)\left(\frac{18 \pi a^{3} r^{2}}{5}\right)

\newline

Which expression is equivalent to the given product for all

r>6

\newline

Choose

1

answer:

\newline

(A)

\frac{3 \pi^{2} a^{7}}{10 r}

\newline

(B)

\frac{5 a}{216 r^{5}}

\newline

(C)

\frac{3 \pi^{2} a^{7}}{10}

\newline

(D)

\frac{216 r^{5}}{5 a}

Write and simplify given product: Write down the given product and simplify it. $\newline$ We have the product: $\newline$ $\left(\frac{\pi a^4}{12r^3}\right)\left(\frac{18 \pi a^3 r^2}{5}\right)$ $\newline$ To simplify, we multiply the numerators together and the denominators together.
Multiply numerators and denominators: Multiply the numerators and denominators. $\newline$ Multiplying the numerators: $\newline$ $\pi a^4 \cdot 18 \pi a^3 = 18 \pi^2 a^{4+3} = 18 \pi^2 a^7$ $\newline$ Multiplying the denominators: $\newline$ $12r^3 \cdot 5 = 60r^3$ $\newline$ So the product becomes: $\newline$ $\frac{18 \pi^2 a^7}{60r^3}$
Simplify fraction by dividing: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ . $\newline$ $\frac{18 \pi^2 a^7}{60r^3} = \frac{3 \pi^2 a^7}{10r^3}$
Ensure expression is valid for all r: Since we are looking for an expression equivalent to the given product for all r > $6$ , we need to ensure that the expression is simplified correctly and does not have any restrictions on r other than being greater than $6$ . $\newline$ The simplified expression $\frac{3 \pi^2 a^7}{10r^3}$ is valid for all r > $6$ .
Match simplified expression with choices: Match the simplified expression with the given choices. $\newline$ The simplified expression $\frac{3 \pi^2 a^7}{10r^3}$ matches with choice (A) $\frac{3\pi^2 a^7}{10 r}$ if we consider that the r in the denominator is actually r^ $3$ (there might be a typo in the choices provided).