Simplify Radicals with Variables 3

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Question 1 of 4

1. Question
Simplify

`sqrt(8a^7)`
- `2a^3 sqrt(2a)`
- `4a^4`
- `2a sqrt(2a)`
- ` sqrt(2a^3)`
Hint

Help Video

Correct

Excellent!

Incorrect

Multiplication Property of Radicals

`sqrt(ab)=sqrt(a) xx sqrt(b)`

First, separate the variable from the constant.

`sqrt(8a^7)` `=` `sqrt(8)`` xx ``sqrt(a^7)` Apply the Multiplication Property

Find factors that are perfect squares for `sqrt(a^7)`

`sqrt(a^7)` `=` `sqrt(a^6 xx a)`

`=` `a^3 sqrta` `a^6` is a perfect square

Simplify the values further

`sqrt8``xx``a^3sqrta` `=` `sqrt(4 xx 2)``xx``a^3sqrta`

`=` `2sqrt2 xx a^3sqrta` `4` is a perfect square

`=` `2a^3 sqrt(2a)`

`2a^3 sqrt(2a)`

Question 2 of 4

Simplify

`root (3)(54y^4)`

`3yroot(3)(2y)`
`2y`
`3yroot(3)(3y)`
`root(3)(2y)`

Question 3 of 4

Simplify

`sqrt(405a^6b^3)`

`9a^3bsqrt(5b)`
`9a^3bsqrt5`
`9a^3`
`sqrt(9a^3b)`

Question 4 of 4

4. Question
Simplify

`root(3)(-8x^6y^(12))`
- `-2x^2y^4`
- `2x^2y^4`
- `y^4`
- `-2x^3y^3`
Hint

Help Video

Correct

Correct!

Incorrect

Multiplication Property of Radicals

`sqrt(ab)=sqrt(a) xx sqrt(b)`

First, separate the variable from the constant.

`root(3)(-8x^6y^(12))` `=` `root(3)(-8)`` xx ``root(3)(x^6y^(12))` Apply the Multiplication Property

Find factors that are perfect cubes for `root (3)(x^6y^(12))`

`=` `root(3)((x^2)^3 (y^4)^3)` Cube root cancels the power

`=` `x^2y^4`

Simplify the values further

`root(3)(-8)`` xx ``x^2y^4` `=` `-2``xx``x^2y^4` `-8` is a perfect cube

`=` `-2x^2y^4`

`-2x^2y^4`

`root(3)(y^4)`	`=`	`root(3)(y^3 xx y)`
	`=`	`y root(3)y`	`y^3` is a perfect cube

`sqrt(a^6b^3)`	`=`	`a^3sqrt(b^3)`	`a^6` is a perfect square
	`=`	`a^3sqrt(b^2 xx b)`
	`=`	`a^3 xx sqrt(b^2) xx sqrtb`	Apply Multiplication Property
	`=`	`a^3b sqrtb`	`b^2` is a perfect square

`sqrt(405)``xx``a^3bsqrtb`	`=`	`sqrt(81 xx 5)``xx``a^3bsqrtb`
	`=`	`9 xx sqrt5 xx a^3b xx sqrtb`	`81` is a perfect square
	`=`	`9a^3b xx sqrt(5b)`	Combine like terms
	`=`	`9a^3bsqrt(5b)`

Simplify Radicals with Variables 3

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Quiz summary

Information

1. Question

Hint

Excellent!

Multiplication Property of Radicals

2. Question

Hint

Keep Going!

Multiplication Property of Radicals

3. Question

Hint

Fantastic!

Multiplication Property of Radicals

4. Question

Hint

Correct!

Multiplication Property of Radicals

Quizzes

`sqrt(a^7)`	`=`	`sqrt(a^6 xx a)`
	`=`	`a^3 sqrta`	`a^6` is a perfect square

`sqrt8``xx``a^3sqrta`	`=`	`sqrt(4 xx 2)``xx``a^3sqrta`
	`=`	`2sqrt2 xx a^3sqrta`	`4` is a perfect square
	`=`	`2a^3 sqrt(2a)`

`root(3)(54)``xx``yroot(3)y`	`=`	`root(3)(27 xx 2)``xx``yroot(3)y`
	`=`	`root(3) (27) xx root(3)(2) xx y root(3) y`	Apply the Multiplication Property
	`=`	`3 xx root(3) 2 xx y root (3) y`	`27` is a perfect cube
	`=`	`3y xx root (3)(2xxy)`	Combine the terms
	`=`	`3yroot(3)(2y)`

	`=`	`root(3)((x^2)^3 (y^4)^3)`	Cube root cancels the power
	`=`	`x^2y^4`

`root(3)(-8)`` xx ``x^2y^4`	`=`	`-2``xx``x^2y^4`	`-8` is a perfect cube
	`=`	`-2x^2y^4`