Simplify Roots of Negative Numbers 2

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Question 1 of 5

Simplify

$\sqrt{- \frac{1}{9}}$ $sqrt(-1/9)$

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

Question 2 of 5

Simplify

$\sqrt{4 - 4 (17)}$ $sqrt(4-4(17))$

1. $- 16 i$ $-16i$
2. $4 i$ $4i$
3. $64 i$ $64i$
4. $8 i$ $8i$

Question 3 of 5

Simplify

$\frac{4 \pm \sqrt{36 - 4 (13)}}{2}$ $(4\+-sqrt(36-4(13)))/2$

1. $4 \pm 2 i$ $4\+-2i$
2. $2 \pm 4 i$ $2\+-4i$
3. $2 \pm 2 i$ $2\+-2i$
4. $4 \pm 4 i$ $4\+-4i$

Question 4 of 5

Simplify

$\frac{2 + \sqrt{4 - 164}}{2}$ $(2+sqrt(4-164))/2$

1. $1 \pm i \sqrt{10}$ $1+-isqrt(10)$
2. $1 \pm 2 i$ $1+-2i$
3. $1 \pm 2 i \sqrt{10}$ $1+-2isqrt(10)$
4. $1 \pm 2 \sqrt{10}$ $1+-2sqrt(10)$

Question 5 of 5

Simplify

$(-6+-sqrt(16-4(16)))/2$

1. $-3+-2isqrt(3)$
2. $3+-2isqrt(3)$
3. $-3+-2i$
4. $3+-2i$

$=$ $=$	$\sqrt{- 1} \times \sqrt{\frac{1}{9}}$ $color(royalblue)(sqrt(-1))timescolor(forestgreen)(sqrt(1/9))$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$i \times \sqrt{\frac{1}{9}}$ $color(royalblue)(i)timescolor(forestgreen)(sqrt(1/9))$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$i \times \frac{1}{\sqrt{9}}$ $itimes1/(sqrt(9)$	Simplify $\sqrt{\frac{1}{9}}$ $sqrt(1/9)$ .
$=$ $=$	$i \times \frac{1}{3}$ $itimes1/3$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$\frac{1}{3} i$ $1/3i$

$=$ $=$	$\sqrt{4 - 4 \times 17}$ $sqrt(4-4timescolor(royalblue)(17)$	Multiply $4$ $4$ and $17$ $17$ .
$=$ $=$	$\sqrt{4 - 68}$ $sqrt(4-color(royalblue)(68)$	Subtract.
$=$ $=$	$\sqrt{- 64}$ $sqrt(-64)$	Remove the negative from the root by using $i$ $i$ .
$=$ $=$	$i \sqrt{64}$ $isqrt(64)$	Simplify $\sqrt{64}$ $sqrt(64)$ .
$=$ $=$	$i \times 8$ $itimes8$	Rearrange the answer so that the $i$ $i$ is on the right-hand side.
$=$ $=$	$8 i$ $8i$

	$\frac{4 \pm \sqrt{36 - 4 (13)}}{2}$ $(4\+-sqrt(36-color(royalblue)(4(13))))/2$	Multiply
$=$ $=$	$\frac{4 \pm \sqrt{36 - 52}}{2}$ $(4\+-sqrt(36-color(royalblue)(52)))/2$	Subtract under the root sign.
$=$ $=$	$\frac{4 \pm \sqrt{- 16}}{2}$ $(4\+-sqrt(-16))/2$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$=$ $=$	$\frac{4 \pm \sqrt{- 1} \times \sqrt{16}}{2}$ $(4\+-sqrt(-1)timessqrt(16))/2$	Replace $\sqrt{- 1}$ $sqrt(-1)$ with $i$ $i$ .
$=$ $=$	$\frac{4 \pm i \times \sqrt{16}}{2}$ $(4\+-itimessqrt(16))/2$	Simplify $\sqrt{16}$ $sqrt(16)$ .
$=$ $=$	$\frac{4 \pm i \times 4}{2}$ $(4\+-itimes4)/2$	Divide both terms in the numerator by $2$ $2$ .
$=$ $=$	$2 \pm i \times 2$ $2\+-itimes 2$	Rearrange the answer so that the $i$ $i$ is on the right-hand side in its term.
$=$ $=$	$2 \pm 2 i$ $2\+-2i$

$=$ $=$	$\frac{2 + \sqrt{4 - 164}}{2}$ $(2+color(royalblue)sqrt(4-164))/2$	Subtract $4 - 164$ $4-164$ .
$=$ $=$	$\frac{2 \pm \sqrt{- 160}}{2}$ $(2+-color(royalblue)sqrt(-160))/2$	Remove the negative root by using $i$ $i$ .
$=$ $=$	$\frac{2 \pm i \sqrt{160}}{2}$ $(2+-icolor(royalblue)sqrt(160))/2$	$\sqrt{160} = 4 \sqrt{10}$ $sqrt(160) = 4sqrt(10)$ .
$=$ $=$	$\frac{2 \pm i 4 \sqrt{10}}{2}$ $(2+-icolor(royalblue)4sqrt(10))/2$	Simplify.
$=$	$1+-2isqrt(10)$

$=$	$(-6+-color(royalblue)sqrt(16-4(16)))/2$	Subtract $16-4(16)$ .
$=$	$(-6+-color(royalblue)sqrt(-48))/2$	Remove the negative root by using $i$ .
$=$	$(-6+-icolor(royalblue)sqrt(48))/2$	$sqrt(48) = 4sqrt(3)$ .
$=$	$(-6+-icolor(royalblue)4sqrt(3))/2$	Simplify.
$=$	$-3+-2isqrt(3)$

Simplify Roots of Negative Numbers 2

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