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Simplify Surds with Variables

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Simplify Surds with Variables 2

Simplify Surds with Variables 2

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Question 1 of 4

1. Question
Simplify

$5 \sqrt{a^{8}}$ $5sqrt(a^8)$
- 1. $5 \sqrt{a^{2}}$ $5 sqrt(a^2)$
- 2. $5 a^{2}$ $5a^2$
- 3. $5 a^{4}$ $5a^4$
- 4. $20$ $20$
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Multiplication Property of Surds

$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$ $sqrt(ab)=sqrt(a) xx sqrt(b)$

Rewrite the square root as an exponent

$\sqrt{a^{8}}$ $sqrt(a^8)$ $=$ $=$ ${(a^{8})}^{\frac{1}{2}}$ $(a^8)^(1/2)$ $x^{\frac{1}{2}}$ $x^(1/2)$ is the same as $\sqrt{x}$ $sqrtx$

$=$ $=$ $a^{4}$ $a^4$ Use the Power Rule to simplify ${(x^{a})}^{b} = x^{a b}$ $(x^a)^b = x^(ab)$

Simplify

$=$ $=$ $5 \times$ $5xx$ $a^{4}$ $a^4$

$=$ $=$ $5 a^{4}$ $5a^4$

$5 a^{4}$ $5a^4$

Question 2 of 4

2. Question

Simplify

\sqrt{8 x^{4}}

$sqrt(8x^4)$

1. $9 \sqrt{2 a}$ $9 sqrt(2a)$
2. $72 \sqrt{a}$ $72 sqrt(a)$
3. $9 \sqrt{a}$ $9 sqrt(a)$
4. $2 x^{2} \sqrt{2}$ $2x^2sqrt2$

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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 x^{4}}

$sqrt(8x^4)$

=

$=$

$\sqrt{8}$ $sqrt8$

\times

$xx$ $\sqrt{x^{4}}$ $sqrt(x^4)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

\sqrt{x^{4}}

$sqrt(x^4)$ as a fractional exponent

$\sqrt{x^{4}}$ $sqrt(x^4)$	$=$ $=$	${(x^{4})}^{\frac{1}{2}}$ $(x^4)^(1/2)$
	$=$ $=$	$x^{2}$ $x^2$	Use the Power Rule to simplify ${(x^{a})}^{b} = x^{a b}$ $(x^a)^(b)=x^(ab)$

Finally, find factors that are perfect squares for

\sqrt{8} \times x^{2}

$sqrt8 xx x^2$

$\color{#007DDC}{\sqrt{8}}\times\color{#9a00c7}{x^{2}}$	$=$ $=$	$\sqrt{4 \times 2}$ $sqrt(4 xx 2)$ $\times$ $xx$ $x^{2}$ $x^2$
	$=$ $=$	$\sqrt{4} \times \sqrt{2} \times x^{2}$ $sqrt4 xx sqrt2 xx x^2$	Apply the Multiplication Property
	$=$ $=$	$2 \times \sqrt{2} \times x^{2}$ $2 xx sqrt2 xx x^2$	$4$ $4$ is a perfect square
	$=$ $=$	$2 x^{2} \sqrt{2}$ $2x^2sqrt2$	Rearrange the expression

2 x^{2} \sqrt{2}

$2x^2sqrt2$

Question 3 of 4

3. Question

Simplify

3 y \sqrt{128 y^{4}}

$3ysqrt(128y^4)$

1. $24 y^{3} \sqrt{2}$ $24y^3sqrt2$
2. $24 y \sqrt{2 y}$ $24ysqrt(2y)$
3. $24 y \sqrt{2}$ $24ysqrt2$
4. $8 y^{3} \sqrt{2}$ $8y^3sqrt2$

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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 in the surd.

\sqrt{128 y^{4}}

$sqrt(128y^4)$

=

$=$

$\sqrt{128}$ $sqrt(128)$

\times

$xx$ $\sqrt{y^{4}}$ $sqrt(y^4)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

\sqrt{y^{4}}

$sqrt(y^4)$ as a fractional exponent

$\sqrt{y^{4}}$ $sqrt(y^4)$	$=$ $=$	${(y^{4})}^{\frac{1}{2}}$ $(y^4)^(1/2)$
	$=$ $=$	$y^{2}$ $y^2$	Use the Power Rule to simplify ${(x^{a})}^{b} = x^{a b}$ $(x^a)^(b)=x^(ab)$

Finally, find factors that are perfect squares for

3 y \times \sqrt{128} \times y^{2}

$3y xx sqrt128 xx y^2$

$3y\times\color{#007DDC}{\sqrt{128}}\times\color{#9a00c7}{y^{2}}$	$=$ $=$	$3 y \times$ $3y xx$ $\sqrt{64 \times 2}$ $sqrt(64 xx 2)$ $\times$ $xx$ $y^{2}$ $y^2$
	$=$ $=$	$3 y \times \sqrt{64} \times \sqrt{2} \times y^{2}$ $3y xx sqrt64 xx sqrt2 xx y^2$	Apply the Multiplication Property
	$=$ $=$	$3 y \times 8 \times \sqrt{2} \times y^{2}$ $3y xx 8 xx sqrt2 xx y^2$	$64$ $64$ is a perfect square
	$=$ $=$	$24 y^{3} \sqrt{2}$ $24y^3sqrt2$	Rearrange the expression

24 y^{3} \sqrt{2}

$24y^3sqrt2$

Question 4 of 4

4. Question

Simplify

\sqrt{9 x^{2} + 24 x y + 16 y^{2}}

$sqrt(9x^2+24xy+16y^2)$

1. $4 y - 3 x$ $4y-3x$
2. $7$ $7$
3. $3 x - 4 y$ $3x-4y$
4. $3 x + 4 y$ $3x+4y$

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Multiplication Property of Surds

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

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

Simplify the surd further

$\sqrt{9 x^{2} + 24 x y + 16 y^{2}}$ $sqrt(9x^2+24xy+16y^2)$	$=$ $=$	$\sqrt{(3 x + 4 y) (3 x + 4 y)}$ $sqrt((3x+4y)(3x+4y))$
	$=$ $=$	$\sqrt{{(3 x + 4 y)}^{2}}$ $sqrt((3x+4y)^2)$	Apply Exponent Rules
	$=$ $=$	$3 x + 4 y$ $3x+4y$	${(3 x + 4 y)}^{2}$ $(3x+4y)^2$ is a perfect square

$3x+4y$

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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