Simplify Surds with Variables 1

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Year 8

Surds

Simplify Surds with Variables

Simplify Surds with Variables 1

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Question 1 of 5

Simplify

8 \sqrt{162 a}

$8sqrt(162a)$

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

Question 2 of 5

Simplify

\sqrt{24 x^{2}}

$sqrt(24x^2)$

1. $2 x \sqrt{6}$ $2xsqrt6$
2. $\sqrt{6}$ $sqrt6$
3. $2 \sqrt{6}$ $2sqrt6$
4. $12$ $12$

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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

Simplify

$sqrt(16x^3)$

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

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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. $9y+7$
- 2. $(y+7)^2$
- 3. $3(y+7)^2$
- 4. $9$
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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. $sqrt(x^3)$
- 2. $x^3$
- 3. $xsqrtx$
- 4. $x^3sqrtx$
Hint

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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

$=$ $x^3sqrtx$

$x^3sqrtx$

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

Simplify Surds with Variables 1

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Quiz summary

Information

1. Question

Hint

Great Work!

Multiplication Property of Surds

2. Question

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Excellent!

Multiplication Property of Surds

3. Question

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Nice Job!

Multiplication Property of Surds

4. Question

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Well Done!

Multiplication Property of Surds

5. Question

Hint

Multiplication Property of Surds

Quizzes

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

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

$sqrt(x^7)$	$=$	$sqrt(x^6) xx sqrtx$	Find the highest even factor of $x^7$
	$=$	$x^3 xx sqrtx$	$sqrt(x^6)$ is a perfect square
	$=$	$x^3sqrtx$