Simplify Roots of Negative Numbers 1

Question 1 of 6

Simplify

$\sqrt{- 36}$ $sqrt(-36)$

1. $3 i$ $3i$
2. $6 i$ $6i$
3. $9 i$ $9i$
4. $12 i$ $12i$

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Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- 36}

$sqrt(-36)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{36}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(36))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{36}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(36))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times (\sqrt{36})$ $itimes(sqrt(36))$	Simplify $\sqrt{36}$ $sqrt(36)$ .
$=$ $=$	$i \times 6$ $itimes6$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$6 i$ $6i$

6 i

$6i$

Question 2 of 6

Simplify

$\sqrt{- 5}$ $sqrt(-5)$

1. $i \sqrt{5}$ $isqrt(5)$
2. $2.5 i$ $2.5i$
3. $5 i$ $5i$
4. $\sqrt{5}$ $sqrt(5)$

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Mute

Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- 5}

$sqrt(-5)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{5}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(5))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{5}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(5))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times \sqrt{5}$ $itimessqrt(5)$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$i \sqrt{5}$ $isqrt(5)$

i \sqrt{5}

$isqrt(5)$

Question 3 of 6

Simplify

$\sqrt{- 48}$ $sqrt(-48)$

1. $4 i \sqrt{3}$ $4isqrt(3)$
2. $\sqrt{3}$ $sqrt(3)$
3. $4 i$ $4i$
4. $8 \sqrt{3}$ $8sqrt(3)$

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Mute

Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- 48}

$sqrt(-48)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{48}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(48))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{48}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(48))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times (\sqrt{48})$ $itimes(sqrt(48))$	Simplify $\sqrt{48} = \sqrt{16} \times \sqrt{3}$ $sqrt(48) = sqrt(16) xx sqrt(3)$ .
$=$ $=$	$i \times (\sqrt{16}) \times (\sqrt{3})$ $itimes(sqrt(16))times(sqrt(3))$	Simplify `sqrt(16).
$=$ $=$	$i \times (4) \times (\sqrt{3})$ $itimes(4)times(sqrt(3))$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$4 i \sqrt{3}$ $4isqrt(3)$

4 i \sqrt{3}

$4isqrt(3)$

Question 4 of 6

Simplify

$\sqrt{- 108}$ $sqrt(-108)$

1. $\sqrt{3}$ $sqrt(3)$
2. $i \sqrt{3}$ $isqrt(3)$
3. $6 \sqrt{3}$ $6sqrt(3)$
4. $6 i \sqrt{3}$ $6isqrt(3)$

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Mute

Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- 108}

$sqrt(-108)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{108}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(108))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{108}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(108))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times \sqrt{108}$ $itimessqrt(108)$	$\sqrt{108} = \sqrt{36} \times \sqrt{3}$ $sqrt(108) = sqrt(36) xx sqrt(3)$ .
$=$ $=$	$i \times (\sqrt{36}) \times (\sqrt{3})$ $itimes(sqrt(36))times(sqrt(3))$	Simplify $\sqrt{36}$ $sqrt(36)$ .
$=$ $=$	$i \times (6) \times (\sqrt{3})$ $itimes(6)times(sqrt(3))$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$6 i \sqrt{3}$ $6isqrt(3)$

6 i \sqrt{3}

$6isqrt(3)$

Question 5 of 6

Simplify

$\sqrt{- 10} \times \sqrt{- 15}$ $sqrt(-10) xx sqrt(-15)$

1. $- 5 \sqrt{6}$ $-5sqrt(6)$
2. $5 i \sqrt{6}$ $5isqrt(6)$
3. $5 \sqrt{6}$ $5sqrt(6)$
4. $\sqrt{6}$ $sqrt(6)$

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Mute

Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- 10}

$sqrt(-10)$ and

\sqrt{- 15}

$sqrt(-15)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{10} \times \sqrt{- 1} \times \sqrt{15}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(10)) xx color(accentorange)(sqrt(-1))timescolor(crimson)(sqrt(15))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{10} \times i \times \sqrt{15}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(10)) xx color(accentorange)(i)timescolor(crimson)(sqrt(15))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times (\sqrt{5} \times \sqrt{2}) \times i \times (\sqrt{5} \times \sqrt{3})$ $itimes(sqrt(5) xx sqrt(2)) xx itimes(sqrt(5) xx sqrt(3))$	$\sqrt{10} = \sqrt{5} \times \sqrt{2}$ $sqrt(10) = sqrt(5) xx sqrt(2)$ and $\sqrt{15} = \sqrt{5} \times \sqrt{3}$ $sqrt(15) = sqrt(5) xx sqrt(3)$ .
$=$ $=$	$- 1 \times 5 \times \sqrt{6}$ $-1times5timessqrt(6)$	Simplify.
$=$ $=$	$- 5 \sqrt{6}$ $-5sqrt(6)$

- 5 \sqrt{6}

$-5sqrt(6)$

Question 6 of 6

Simplify

$\sqrt{- \frac{1}{4}}$ $sqrt(-1/4)$

1. $4 i$ $4i$
2. $\frac{1}{2} i$ $1/2i$
3. $2 i$ $2i$
4. $\frac{1}{4} i$ $1/4i$

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Mute

Imaginary numbers have the properties

i = \sqrt{- 1}

$i=sqrt(-1)$ or

i^{2} = - 1

$i^2=-1$ .

Remove the negative from the root in

\sqrt{- \frac{1}{4}}

$sqrt(-1/4)$ and then simplify.

$=$ $=$	$\sqrt{- 1} \times \sqrt{\frac{1}{4}}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(1/4))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{\frac{1}{4}}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(1/4))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times \frac{1}{\sqrt{4}}$ $itimes1/(sqrt(4)$	Simplify $\sqrt{\frac{1}{4}}$ $sqrt(1/4)$ .
$=$ $=$	$i \times \frac{1}{2}$ $itimes1/2$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$\frac{1}{2} i$ $1/2i$

\frac{1}{2} i

$1/2i$

Simplify Roots of Negative Numbers 1

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