Simplify Surds 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

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Correct

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Incorrect

Multiplication Property of Surds

`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 Surds

`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 Surds with Variables 3

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Quiz summary

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1. Question

Hint

Excellent!

Multiplication Property of Surds

2. Question

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Keep Going!

Multiplication Property of Surds

3. Question

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Fantastic!

Multiplication Property of Surds

4. Question

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Correct!

Multiplication Property of Surds

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`