Question

When x > 0 and y > 0, what expression is equivalent to √180x^9y^16 in simplest form?

Answer 1

`6x^4y^8√(5x)`

Explanation:

`\textsf{When $x > 0$ and $y > 0$, we want to find the expression that is equivalent to}`
`\sqrt{180x^9y^(16)}.`

`\textsf{First, apply the radical rule:} \quad √(ab)=√(a)√(b)`

`√(180)√(x^9)\sqrt{y^(16)}`

`\textsf{Rewrite $x^9$ as $x^(8+1)$:}`

`√(180)\sqrt{x^((8+1))}\sqrt{y^(16)}`

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

`√(180)\sqrt{x^(8)\cdot x^1}\sqrt{y^(16)}`

`√(180)\sqrt{x^(8)}√(x)\sqrt{y^(16)}`

`\textsf{Apply\:the\:radical\:rule:\:}\sqrt[n]{a^m}=a^{(m)/(n)},\:\quad a\geq 0`

`√(180)\;x^{(8)/(2)}√(x)\;y^{(16)/(2)}`

`√(180)\;x^4√(x)\;y^8`

`√(180)√(x)\;x^4\;y^8`

`\textsf{Rewrite $180$ as $(6^2 \cdot 5)$:}`

`√(6^2 \cdot 5)√(x)\;x^4\;y^8`

`\textsf{Apply the radical rule:} \quad √(ab)=√(a)√(b)`

`√(6^2) √(5)√(x)\;x^4\;y^8`

`\textsf{Apply the radical rule:} \quad √(a^2)=a, \quad a \geq 0`

`6 √(5)√(x)\;x^4\;y^8`

`\textsf{Apply the radical rule:} \quad √(a)√(b)=√(ab)`

`6 √(5x)\;x^4\;y^8`

`\textsf{Rearrange:}`

`6x^4y^8√(5x)`

`\textsf{Therefore, when $x > 0$ and $y > 0$, the expression that is equivalent to}`

`\sqrt{180x^9y^(16)}\;\textsf{is}\;\;\boxed{6x^4y^8√(5x)}\:.`