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Simplify Surds with Variables

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Simplify Surds with Variables 1

Simplify Surds with Variables 1

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Question 1 of 5

1. Question

Simplify

$8sqrt(162a)$

1. $9 sqrt(2a)$
2. $9 sqrt(a)$
3. $72 sqrt(2a)$
4. $72 sqrt(a)$

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Multiplication Property of Surds

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

Find factors that are perfect squares for

$sqrt(162a)$

$8 sqrt(162a)$	$=$	$8 sqrt(81 xx 2 xx a)$	Factor by finding the greatest perfect square of $162$
	$=$	$8 xx sqrt81 xx sqrt2 xx sqrta$	Apply the Multiplication Property
	$=$	$8 xx 9 xx sqrt2 xx sqrta$	$81$ is a perfect square
	$=$	$72 xx sqrt2 xx sqrta$
	$=$	$72 sqrt(2a)$

$72 sqrt(2a)$

Question 2 of 5

2. Question

Simplify

$sqrt(24x^2)$

1. $12$
2. $2sqrt6$
3. $sqrt6$
4. $2xsqrt6$

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Multiplication Property of Surds

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

First, separate the variable from the constant.

$sqrt(24x^2)$

$=$

$sqrt(24)$

$xx$ $sqrt(x^2)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

$sqrt(x^2)$ as a fractional exponent

$sqrt(x^2)$	$=$	$(x^2)^(1/2)$
	$=$	$x$	Use the Power Rule to simplify $(x^a)^(b)=x^(ab)$

Finally, find factors that are perfect squares for

$sqrt24 xx x$

$\color{#007DDC}{\sqrt{24}}\times\color{#9a00c7}{x}$	$=$	$sqrt(4 xx 6)$ $xx$ $x$
	$=$	$sqrt4 xx sqrt6 xx x$	Apply the Multiplication Property
	$=$	$2 xx sqrt6 xx x$	$4$ is a perfect square
	$=$	$2xsqrt6$	Rearrange the expression

$2xsqrt6$

Question 3 of 5

3. Question

Simplify

$sqrt(16x^3)$

1. $4xsqrtx$
2. $4sqrtx$
3. $2xsqrtx$
4. $4x$

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Multiplication Property of Surds

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

First, separate the variable from the constant.

$sqrt(16x^3)$

$=$

$sqrt(16)$

$xx$ $sqrt(x^3)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

$sqrt(x^3)$ as a fractional exponent

$sqrt(x^3)$	$=$	$sqrt(x^2 xx x)$
$sqrt(x^3)$	$=$	$sqrt(x^2) xx sqrtx$	Apply the Multiplication Property
	$=$	$x xx sqrtx$	Use the Power Rule to simplify $(x^a)^(b)=x^(ab)$
	$=$	$xsqrtx$

Finally, find factors that are perfect squares for

$sqrt16 xx xsqrtx$

$\color{#007DDC}{\sqrt{16}}\times\color{#9a00c7}{x\sqrt{x}}$	$=$	$4$ $xx$ $x$
	$=$	$4xsqrtx$	Rearrange the expression

$4xsqrtx$

Question 4 of 5

4. Question
Simplify

$sqrt(9(y+7)^4)$
- 1. $9$
- 2. $9y+7$
- 3. $(y+7)^2$
- 4. $3(y+7)^2$
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Multiplication Property of Surds

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

First, separate the variable from the constant.

$sqrt(9(y+7)^4)$ $=$ $sqrt(9)$ $xx$ $sqrt((y+7)^4)$ Apply the Multiplication Property

Next, get the factor of $sqrt((y+7)^4)$ and simplify

$sqrt((y+7)^4)$ $=$ $sqrt((y+7)^2 xx (y+7)^2)$

$=$ $(y+7)^2$ $(y+7)^4$ is a perfect square

Finally, find factors that are perfect squares for $sqrt9 xx (y+7)^2$

$\color{#007DDC}{\sqrt{9}}\times\color{#9a00c7}{(y+7)^2}$ $=$ $3$ $xx$ $(y+7)^2$

$=$ $3(y+7)^2$

$3(y+7)^2$
Question 5 of 5

5. Question
Simplify

$sqrt(x^7)$
- 1. $x^3sqrtx$
- 2. $x^3$
- 3. $sqrt(x^3)$
- 4. $xsqrtx$
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Multiplication Property of Surds

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

Find factors that are perfect squares for $sqrt(x^7)$

$sqrt(x^7)$ $=$ $sqrt(x^6) xx sqrtx$ Find the highest even factor of $x^7$

$=$ $x^3 xx sqrtx$ $sqrt(x^6)$ is a perfect square

$=$ $x^3sqrtx$

$x^3sqrtx$

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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