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Simplify Surds with Variables

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Simplify Surds with Variables 3

Simplify Surds with Variables 3

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Question 1 of 4

1. Question

Simplify

\sqrt{8 a^{7}}

$sqrt(8a^7)$

1. $2 a^{3} \sqrt{2 a}$ $2a^3 sqrt(2a)$
2. $2 a \sqrt{2 a}$ $2a sqrt(2a)$
3. $\sqrt{2 a^{3}}$ $sqrt(2a^3)$
4. $4 a^{4}$ $4a^4$

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Multiplication Property of Surds

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

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

First, separate the variable from the constant.

\sqrt{8 a^{7}}

$sqrt(8a^7)$

=

$=$

$\sqrt{8}$ $sqrt(8)$

\times

$xx$ $\sqrt{a^{7}}$ $sqrt(a^7)$

Apply the Multiplication Property

Find factors that are perfect squares for

\sqrt{a^{7}}

$sqrt(a^7)$

$\sqrt{a^{7}}$ $sqrt(a^7)$	$=$ $=$	$\sqrt{a^{6} \times a}$ $sqrt(a^6 xx a)$
	$=$ $=$	$a^{3} \sqrt{a}$ $a^3 sqrta$	$a^{6}$ $a^6$ is a perfect square

Simplify the values further

$\sqrt{8}$ $sqrt8$ $\times$ $xx$ $a^{3} \sqrt{a}$ $a^3sqrta$	$=$ $=$	$\sqrt{4 \times 2}$ $sqrt(4 xx 2)$ $\times$ $xx$ $a^{3} \sqrt{a}$ $a^3sqrta$
	$=$ $=$	$2 \sqrt{2} \times a^{3} \sqrt{a}$ $2sqrt2 xx a^3sqrta$	$4$ $4$ is a perfect square
	$=$ $=$	$2 a^{3} \sqrt{2 a}$ $2a^3 sqrt(2a)$

2 a^{3} \sqrt{2 a}

$2a^3 sqrt(2a)$

Question 2 of 4

2. Question

Simplify

\sqrt[3]{54 y^{4}}

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

1. $\sqrt[3]{2 y}$ $root(3)(2y)$
2. $3 y \sqrt[3]{3 y}$ $3yroot(3)(3y)$
3. $2 y$ $2y$
4. $3 y \sqrt[3]{2 y}$ $3yroot(3)(2y)$

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Multiplication Property of Surds

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

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

First, separate the variable from the constant.

\sqrt[3]{54 y^{4}}

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

=

$=$

$\sqrt[3]{54}$ $root (3)(54)$

\times

$xx$ $\sqrt[3]{y^{4}}$ $root (3)(y^4)$

Apply the Multiplication Property

Find factors that are perfect cubes for

\sqrt[3]{y^{4}}

$root (3)(y^4)$

$\sqrt[3]{y^{4}}$ $root(3)(y^4)$	$=$ $=$	$\sqrt[3]{y^{3} \times y}$ $root(3)(y^3 xx y)$
	$=$ $=$	$y \sqrt[3]{y}$ $y root(3)y$	$y^{3}$ $y^3$ is a perfect cube

Simplify the values further

$\sqrt[3]{54}$ $root(3)(54)$ $\times$ $xx$ $y \sqrt[3]{y}$ $yroot(3)y$	$=$ $=$	$\sqrt[3]{27 \times 2}$ $root(3)(27 xx 2)$ $\times$ $xx$ $y \sqrt[3]{y}$ $yroot(3)y$
	$=$ $=$	$\sqrt[3]{27} \times \sqrt[3]{2} \times y \sqrt[3]{y}$ $root(3) (27) xx root(3)(2) xx y root(3) y$	Apply the Multiplication Property
	$=$ $=$	$3 \times \sqrt[3]{2} \times y \sqrt[3]{y}$ $3 xx root(3) 2 xx y root (3) y$	$27$ $27$ is a perfect cube
	$=$ $=$	$3 y \times \sqrt[3]{2 \times y}$ $3y xx root (3)(2xxy)$	Combine the terms
	$=$ $=$	$3 y \sqrt[3]{2 y}$ $3yroot(3)(2y)$

3 y \sqrt[3]{2 y}

$3yroot(3)(2y)$

Question 3 of 4

3. Question

Simplify

\sqrt{405 a^{6} b^{3}}

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

1. $9 a^{3}$ $9a^3$
2. $\sqrt{9 a^{3} b}$ $sqrt(9a^3b)$
3. $9 a^{3} b \sqrt{5}$ $9a^3bsqrt5$
4. $9 a^{3} b \sqrt{5 b}$ $9a^3bsqrt(5b)$

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Chapters

Chapters

Multiplication Property of Surds

\sqrt{a b} = \sqrt{a} \times \sqrt{b}

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

First, separate the variable from the constant.

\sqrt{405 a^{6} b^{3}}

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

=

$=$

$\sqrt{405}$ $sqrt(405)$

\times

$xx$ $\sqrt{a^{6} b^{3}}$ $sqrt(a^6b^3)$

Apply the Multiplication Property

Find factors that are perfect squares for

\sqrt{a^{6} b^{3}}

$sqrt(a^6b^3)$

$\sqrt{a^{6} b^{3}}$ $sqrt(a^6b^3)$	$=$ $=$	$a^{3} \sqrt{b^{3}}$ $a^3sqrt(b^3)$	$a^{6}$ $a^6$ is a perfect square
	$=$ $=$	$a^{3} \sqrt{b^{2} \times b}$ $a^3sqrt(b^2 xx b)$
	$=$ $=$	$a^{3} \times \sqrt{b^{2}} \times \sqrt{b}$ $a^3 xx sqrt(b^2) xx sqrtb$	Apply Multiplication Property
	$=$ $=$	$a^{3} b \sqrt{b}$ $a^3b sqrtb$	$b^{2}$ $b^2$ is a perfect square

Simplify the values further

$\sqrt{405}$ $sqrt(405)$ $\times$ $xx$ $a^{3} b \sqrt{b}$ $a^3bsqrtb$	$=$	$sqrt(81 xx 5)$ $xx$ $a^3bsqrtb$
	$=$	$9 xx sqrt5 xx a^3b xx sqrtb$	$81$ is a perfect square
	$=$	$9a^3b xx sqrt(5b)$	Combine like terms
	$=$	$9a^3bsqrt(5b)$

$9a^3bsqrt(5b)$

Question 4 of 4

4. Question
Simplify

$root(3)(-8x^6y^(12))$
- 1. $y^4$
- 2. $-2x^3y^3$
- 3. $-2x^2y^4$
- 4. $2x^2y^4$
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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

$=$ $-2x^2y^4$

$-2x^2y^4$

Quizzes

Simplify Square Roots 1
Simplify Square Roots 2
Simplify Square Roots 3
Simplify Square Roots 4
Simplify Surds with Variables 1
Simplify Surds with Variables 2
Simplify Surds with Variables 3
Rewriting Entire and Mixed Surds 1
Rewriting Entire and Mixed Surds 2
Add and Subtract Surd Expressions (Basic) 1
Add and Subtract Surd Expressions (Basic) 2
Add and Subtract Surd Expressions (Basic) 3
Add and Subtract Surd Expressions 1
Add and Subtract Surd Expressions 2
Add and Subtract Surd Expressions 3
Multiply Surd Expressions 1
Multiply Surd Expressions 2
Multiply Surd Expressions 3
Multiply Surd Expressions 4
Divide Surd Expressions 1
Divide Surd Expressions 2
Divide Surd Expressions 3
Multiply and Divide Surd Expressions
Simplify Surd Expressions using the Distributive Property 1
Simplify Surd Expressions using the Distributive Property 2
Simplify Surd Expressions using the Distributive Property 3
Simplify Binomial Surd Expressions using the FOIL Method 1
Simplify Binomial Surd Expressions using the FOIL Method 2
Rationalising the Denominator 1
Rationalising the Denominator 2
Rationalising the Denominator 3
Rationalising the Denominator 4
Rationalising the Denominator using Conjugates

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