Question

Which expression is equivalent to RootIndex 3 StartRoot StartFraction 75 a Superscript 7 Baseline b Superscript 4 Baseline Over 40 a Superscript 13 Baseline c Superscript 9 Baseline EndFraction EndRoot? Assume a not-equals 0 and c not-equals 0.

A. StartFraction a cubed b (RootIndex 3 StartRoot 15 b squared EndRoot) Over 2 c cubed EndFraction
B. StartFraction b (RootIndex 3 StartRoot 15 b EndRoot) Over 2 a squared c cubed EndFraction
C. StartFraction a cubed b (RootIndex 3 StartRoot 15 b squared EndRoot) Over 6 c cubed EndFraction
D. StartFraction b (RootIndex 3 StartRoot 15 b EndRoot) Over 2 a c EndFraction

Answer 1

B on edge2020

Answer 2

B.
`(b)/(2a^(2)c^3)\sqrt[3]{15b}`

Explanation:

Given:

The expression to simplify is given as:

`\sqrt[3]{(75a^7b^4)/(40a^(13)c^9)}`

Use the exponent property
`(a^m)/(a^n)=a^(m-n)`

`(a^7)/(a^(13))=a^(7-13)=a^(-6)`

Use the exponent property
`(a^m)^n=a^(m* n)`

`a^(-6)=a^(-2* 3)=(a^(-2))^3`

`b^4=b* b^3\\c^(9)=(c^3)^3`

Reducing
`(75)/(40)` to simplest form, we get:

`(5* 5* 3)/(2^3* 5)=(15)/(2^3)`

Therefore, expression becomes:

`\sqrt[3]{(15(a^(-2))^3* b* b^3)/(2^3(c^3)^3)}`

Use the cubic root property:

`\sqrt[3]{x^3} =x`. Thus, the expression becomes:

`(a^(-2)b)/(2c^3)\sqrt[3]{15b}`

Using the exponent property
`a^(-m)=(1)/(a^m)`

`a^(-2)=(1)/(a^2)`

So, the final expression is:

`(b)/(2a^(2)c^3)\sqrt[3]{15b}`

Therefore, the correct option is option B.