Question

How do you simplify `(\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }`?

Answer 1

Here's how it's done

Explanation:

Cancel common factor `x`
`=x^6y^3`
`\=(x^6y^3\)^{\frac{1}{3}}`

Apply the exponent rule
`=\(x^6\)^{\frac{1}{3}}\(y^3\)^{\frac{1}{3}}`

Final solution
`=x^2y`

Answer 2

`" "`
`color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y`

Explanation:

`" "`
Given the expression: `color(red)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }`

`"Exponents formula required: "`

`color(blue)( a^m/a^n = a^(m-n)`

`color(blue)( a^m*a^n = a^(m+n)`

`color(blue)(( a^m)^(1/n) = a^(m/n)`

`color(green)("Step 1:"`

Rewrite the given expression: `color(brown)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }` as

`[[(x^7*y^3)^(1/3)]]/[[x^(1/3)]`

Rewrite as:

`[(x^7)^(1/3)*(y^3)^(1/3)]/x^(1/3)`

`color(green)("Step 2:"`

Use: `color(blue)(( a^m)^(1/n) = a^(m/n)`

Simplify:

`(x^(7/3)*y^(3/3))/x^(1/3)`

`color(green)("Step 3:"`

`(x^(7/3)*y^1)/x^(1/3)`

`color(green)("Step 4:"`

Use: `color(blue)( a^m/a^n = a^(m-n)`

`[x^(7/3)-x^(1/3)]*y`

`x^(6/3)*y`

`rArr x^2*y`

Hence,

`color(blue)((\frac { x ^ { 7} y ^ { 3} } { x } ) ^ { \frac { 1} { 3} }=x^2y`

Hope it helps.