How do I simplify this?

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How do I simplify this?

$\sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}}$ $sqrt((2)/3)+sqrt(3/2)+2sqrt(1/24)$

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Answer 1 · 2018-04-23T23:42:38

It simplifies down to $\sqrt{6}$ $sqrt6$ .

Explanation:

Split the radicals of fractions into a fraction of radicals:

$= \sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}}$ $color(white)=sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + 2 \cdot \frac{\sqrt{1}}{\sqrt{24}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+2*sqrt1/sqrt24$

Simplify a little:

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + 2 \cdot \frac{1}{\sqrt{24}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+2*1/sqrt24$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2}{\sqrt{24}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+2/sqrt24$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2}{2 \sqrt{6}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+2/(2sqrt6)$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{1}{\sqrt{6}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6$

Rationalize the denominator or each fraction:

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$ $=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6color(red)(*sqrt6/sqrt6)$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{\sqrt{6}}{6}$ $=sqrt2/sqrt3+sqrt3/sqrt2+sqrt6/6$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} + \frac{\sqrt{6}}{6}$ $=sqrt2/sqrt3+sqrt3/sqrt2color(red)(*sqrt2/sqrt2)+sqrt6/6$

$= \frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{6}}{2} + \frac{\sqrt{6}}{6}$ $=sqrt2/sqrt3+sqrt6/2+sqrt6/6$

$= \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} + \frac{\sqrt{6}}{2} + \frac{\sqrt{6}}{6}$ $=sqrt2/sqrt3color(red)(*sqrt3/sqrt3)+sqrt6/2+sqrt6/6$

$= \frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{\sqrt{6}}{6}$ $=sqrt6/3+sqrt6/2+sqrt6/6$

Get a common denominator and simplify:

$= \frac{2 \sqrt{6}}{6} + \frac{3 \sqrt{6}}{6} + \frac{\sqrt{6}}{6}$ $=(2sqrt6)/6+(3sqrt6)/6+sqrt6/6$

$= \frac{2 \sqrt{6} + 3 \sqrt{6} + \sqrt{6}}{6}$ $=(2sqrt6+3sqrt6+sqrt6)/6$

$= \frac{6 \sqrt{6}}{6}$ $=(6sqrt6)/6$

$= \sqrt{6}$ $=sqrt6$

That's the simplification. You can check this using a calculator:

![https://www.desmos.com/calculator]( useruploads.socratic.org )

Hope this helped!

Answer 2 · 2018-04-23T23:55:30

$\sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}} = \sqrt{6}$ $sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)=color(blue)(sqrt6$

Explanation:

Simplify:

$\sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}}$ $sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)$

Apply rule $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ $sqrt(a/b)=sqrta/sqrtb$ .

$\frac{\sqrt{2}}{\sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $sqrt2/sqrt3+sqrt3/sqrt2+(2sqrt1)/sqrt24$

Rationalize the denominator in the first term.

$\frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $(sqrt2sqrt3)/(sqrt3sqrt3)+sqrt3/sqrt2+(2sqrt1)/sqrt24$

Apply rule $\sqrt{a} \sqrt{a} = a$ $sqrtasqrta=a$ .

$\frac{\sqrt{2} \sqrt{3}}{3} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $(sqrt2sqrt3)/3+sqrt3/sqrt2+(2sqrt1)/sqrt24$

Apply rule $\sqrt{a} \sqrt{b} = \sqrt{a b}$ $sqrtasqrtb=sqrt(ab)$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{3}}{\sqrt{2}} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $sqrt6/3+sqrt3/sqrt2+(2sqrt1)/sqrt24$

Rationalize the denominator in the second term.

$\frac{\sqrt{6}}{3} + \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $sqrt6/3+(sqrt3sqrt2)/(sqrt2sqrt2)+(2sqrt1)/sqrt24$

Apply rule $\sqrt{a} \sqrt{a} = a$ $sqrtasqrta=a$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{3} \sqrt{2}}{2} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $sqrt6/3+(sqrt3sqrt2)/2+(2sqrt1)/sqrt24$

Apply rule $\sqrt{a} \sqrt{b} = \sqrt{a} b$ $sqrtasqrtb=sqrtab$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \sqrt{1}}{\sqrt{24}}$ $sqrt6/3+sqrt6/2+(2sqrt1)/sqrt24$

Rationalize the denominator in the last term.

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \sqrt{1} \sqrt{24}}{\sqrt{24} \sqrt{24}}$ $sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/(sqrt24sqrt24)$

Apply rule $\sqrt{a} \sqrt{a} = a$ $sqrtasqrta=a$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \sqrt{1} \sqrt{24}}{24}$ $sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/24$

Apply rule $\sqrt{a} \sqrt{b} = \sqrt{a b}$ $sqrtasqrtb=sqrt(ab)$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \sqrt{24}}{24}$ $sqrt6/3+sqrt6/2+(2sqrt24)/24$

Prime factorize $\sqrt{24}$ $sqrt24$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \sqrt{2^{2} \times 2 \times 3}}{24}$ $sqrt6/3+sqrt6/2+(2sqrt(2^2xx2xx3))/24$

Apply rule $\sqrt{a^{2}} = a$ $sqrt(a^2)=a$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{2 \times 2 \sqrt{6}}{24}$ $sqrt6/3+sqrt6/2+(2xx2sqrt6)/24$

Simplify.

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{4 \sqrt{6}}{24}$ $sqrt6/3+sqrt6/2+(4sqrt6)/24$

Simplify $\frac{4 \sqrt{6}}{24}$ $(4sqrt6)/24$ to $\frac{\sqrt{6}}{6}$ $sqrt6/6$ .

$\frac{\sqrt{6}}{3} + \frac{\sqrt{6}}{2} + \frac{\sqrt{6}}{6}$ $sqrt6/3+sqrt6/2+sqrt6/6$

The least common denominator (LCD) is $6$ $6$ .

Multiply the first two terms by a fractional form of $1$ $1$ so that each has a denominator of $6$ $6$ .

$\frac{\sqrt{6}}{3} \times \frac{2}{2} + \frac{\sqrt{6}}{2} \times \frac{3}{3} + \frac{\sqrt{6}}{6}$ $sqrt6/3xxcolor(magenta)2/color(magenta)2+sqrt6/2xxcolor(teal)3/color(teal)3+sqrt6/6$

Simplify.

$\frac{2 \sqrt{6}}{6} + \frac{3 \sqrt{6}}{6} + \frac{\sqrt{6}}{6}$ $(2sqrt6)/6+(3sqrt6)/6+sqrt6/6$

Combine numerators.

$\frac{(2 \sqrt{6} + 3 \sqrt{6} + \sqrt{6})}{6}$ $((2sqrt6+3sqrt6+sqrt6))/6$

Simplify.

$\frac{6 \sqrt{6}}{6}$ $(6sqrt6)/6$

Simplify.

$\sqrt{6}$ $sqrt6$

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How do I simplify this?

$\sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}}$ $sqrt((2)/3)+sqrt(3/2)+2sqrt(1/24)$

2 Answers

Explanation:

Explanation:

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$\sqrt{\frac{2}{3}} + \sqrt{\frac{3}{2}} + 2 \sqrt{\frac{1}{24}}$ $sqrt((2)/3)+sqrt(3/2)+2sqrt(1/24)$