Question

Simplify the equation

Answer 1

`a \sqrt[5]{b}`

Explanation:

Given expression:

`\sqrt[5]{a^3b^2} * \sqrt[5]{a^2b^(-1)}`

`\textsf{Apply exponent rule} \quad \sqrt[n]{a}=a^{(1)/(n)}:`

`(a^3b^2)^{(1)/(5)} * (a^2b^(-1))^{(1)/(5)}`

As the exponents are the same,

`\textsf{apply exponent rule} \quad a^n \cdot c^n=(a \cdot c)^n:`

`(a^3b^2a^2b^(-1))^{(1)/(5)}`

Collect like terms:

`(a^3a^2b^2b^(-1))^{(1)/(5)}`

`\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^(b+c):`

`(a^((3+2))b^((2-1)))^{(1)/(5)}`

Simplify the exponents:

`(a^5b^1)^{(1)/(5)}`

`\textsf{Apply exponent rule} \quad (a^bc^d)^n=(a^b)^n \cdot (c^d)^n:`

`(a^5)^{(1)/(5)} \cdot (b^1)^{(1)/(5)}`

`\textsf{Apply exponent rule} \quad (a^b)^c=a^(bc):`

`a^{(5)/(5)} \cdot b^{(1)/(5)}`

Simplify:

`a^1 \cdot b^{(1)/(5)}`

`a \cdot b^{(1)/(5)}`

`\textsf{Apply exponent rule} \quad a^{(1)/(n)}=\sqrt[n]{a}:`

`a \sqrt[5]{b}`

Answer 2

`a \sqrt[5]{b}`

Explanation:

solution Given:

`\sqrt[5]{ {a}^(3) {b}^(2) } * \sqrt[5]{ {a}^(2){b }^( - 1) }`

since i indices rule if the power is same it can be multiplied or divided.

`\sqrt[5]{ {a}^(3) {b}^(2) * {a}^(2) {b}^( - 1) }`

since power is added of like terms in multiplication

`\sqrt[5]{ {a}^(3 + 2) {b}^(2 - 1) }`

again simplifying

`\sqrt[5]{ {a}^(5) b}`

by using indices formula

`\sqrt[x]{ {a}^(y) } = {a}^{ (y)/(x) }`

we get

`a \sqrt[5]{b}`