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

Question 2 of 5

Simplify

\sqrt{24 x^{2}}

$sqrt(24x^2)$

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

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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{24 x^{2}}

$sqrt(24x^2)$

=

$=$

$\sqrt{24}$ $sqrt(24)$

\times

$xx$ $\sqrt{x^{2}}$ $sqrt(x^2)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

\sqrt{x^{2}}

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

$\sqrt{x^{2}}$ $sqrt(x^2)$	$=$ $=$	${(x^{2})}^{\frac{1}{2}}$ $(x^2)^(1/2)$
	$=$ $=$	$x$ $x$	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{24} \times x

$sqrt24 xx x$

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

2 x \sqrt{6}

$2xsqrt6$

Question 3 of 5

Simplify

\sqrt{16 x^{3}}

$sqrt(16x^3)$

1. $2 x \sqrt{x}$ $2xsqrtx$
2. $4 x$ $4x$
3. $4 \sqrt{x}$ $4sqrtx$
4. $4 x \sqrt{x}$ $4xsqrtx$

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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{16 x^{3}}

$sqrt(16x^3)$

=

$=$

$\sqrt{16}$ $sqrt(16)$

\times

$xx$ $\sqrt{x^{3}}$ $sqrt(x^3)$

Apply the Multiplication Property

Next, simplify the variable by rewriting the square root in

\sqrt{x^{3}}

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

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

Finally, find factors that are perfect squares for

\sqrt{16} \times x \sqrt{x}

$sqrt16 xx xsqrtx$

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

4 x \sqrt{x}

$4xsqrtx$

Question 4 of 5

4. Question
Simplify

$\sqrt{9 {(y + 7)}^{4}}$ $sqrt(9(y+7)^4)$
- 1. $9$ $9$
- 2. $3 {(y + 7)}^{2}$ $3(y+7)^2$
- 3. $9 y + 7$ $9y+7$
- 4. ${(y + 7)}^{2}$ $(y+7)^2$
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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.

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

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

$\sqrt{{(y + 7)}^{4}}$ $sqrt((y+7)^4)$ $=$ $=$ $\sqrt{{(y + 7)}^{2} \times {(y + 7)}^{2}}$ $sqrt((y+7)^2 xx (y+7)^2)$

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

Finally, find factors that are perfect squares for $\sqrt{9} \times {(y + 7)}^{2}$ $sqrt9 xx (y+7)^2$

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

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

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

5. Question
Simplify

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

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

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

$=$ $x^3sqrtx$

$x^3sqrtx$

Simplify Surds with Variables 1

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Quiz summary

Information

1. Question

Hint

Great Work!

Multiplication Property of Surds

2. Question

Hint

Excellent!

Multiplication Property of Surds

3. Question

Hint

Nice Job!

Multiplication Property of Surds

4. Question

Hint

Well Done!

Multiplication Property of Surds

5. Question

Hint

Multiplication Property of Surds

Quizzes

$8 \sqrt{162 a}$ $8 sqrt(162a)$	$=$ $=$	$8 \sqrt{81 \times 2 \times a}$ $8 sqrt(81 xx 2 xx a)$	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
	$=$ $=$	$72 \times \sqrt{2} \times \sqrt{a}$ $72 xx sqrt2 xx sqrta$
	$=$ $=$	$72 \sqrt{2 a}$ $72 sqrt(2a)$

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

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

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