Question

Simplify the expression and rewrite in radical form.

(Equation + multiple choice options in attached image)

Answer 1

Third one:

`\sf 5 x^2y^3 \sqrt[3]{x^2}`

Explanation:

Simplify the expression:

`\sf 5 \sqrt[3]{x} \cdot x^{(7)/(3)} \cdot y^{(7)/(4)} \cdot \sqrt[4]{y^5}`

Convert cube root in terms of power.

`\sf 5 \cdot x^{(1)/(3)}\cdot x^{(7)/(3) }\cdot y^{(7)/(4) }\cdot y^{(5)/(4)}`

Simplify the expression using the product of powers rule.

`\sf 5\cdot x^{\left((1)/(3) + (7)/(3) \right)} y^{\left((7)/(4)+(5)/(4)\right)}`

Simplify the expression of power.

`\sf 5\cdot x^{(8)/(3) }\cdot y^3`

Convert the power of x in terms of cube root.

`\sf 5\cdot \sqrt[3]{x^8} \cdot y^3`

Simplify the expression using the cube root rule.

`\sf 5\cdot \sqrt[3]{x^8} \cdot y^3`

`\sf 5\cdot \sqrt[3]{x^6 \cdot x^2} \cdot y^3`

`\sf 5\cdot x^2 \cdot \sqrt[3]{x^2} \cdot y^3`

So, the answer is:

`\sf 5 x^2y^3 \sqrt[3]{x^2}`

`\hrulefill`

Note:

Power rule:

`\sf a^n \cdot a^m = a^((n+m))`

Product rule:

`\sf (a\cdot b)^n = a^n \cdot b^n`