How do I simplify this?

sqrt((2)/3)+sqrt(3/2)+2sqrt(1/24)

2 Answers
Apr 23, 2018

It simplifies down to sqrt6.

Explanation:

Split the radicals of fractions into a fraction of radicals:

color(white)=sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)

=sqrt2/sqrt3+sqrt3/sqrt2+2*sqrt1/sqrt24

Simplify a little:

=sqrt2/sqrt3+sqrt3/sqrt2+2*1/sqrt24

=sqrt2/sqrt3+sqrt3/sqrt2+2/sqrt24

=sqrt2/sqrt3+sqrt3/sqrt2+2/(2sqrt6)

=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6

Rationalize the denominator or each fraction:

=sqrt2/sqrt3+sqrt3/sqrt2+1/sqrt6color(red)(*sqrt6/sqrt6)

=sqrt2/sqrt3+sqrt3/sqrt2+sqrt6/6

=sqrt2/sqrt3+sqrt3/sqrt2color(red)(*sqrt2/sqrt2)+sqrt6/6

=sqrt2/sqrt3+sqrt6/2+sqrt6/6

=sqrt2/sqrt3color(red)(*sqrt3/sqrt3)+sqrt6/2+sqrt6/6

=sqrt6/3+sqrt6/2+sqrt6/6

Get a common denominator and simplify:

=(2sqrt6)/6+(3sqrt6)/6+sqrt6/6

=(2sqrt6+3sqrt6+sqrt6)/6

=(6sqrt6)/6

=sqrt6

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

https://www.desmos.com/calculatorhttps://www.desmos.com/calculator

Hope this helped!

Apr 23, 2018

sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)=color(blue)(sqrt6

Explanation:

Simplify:

sqrt(2/3)+sqrt(3/2)+2sqrt(1/24)

Apply rule sqrt(a/b)=sqrta/sqrtb.

sqrt2/sqrt3+sqrt3/sqrt2+(2sqrt1)/sqrt24

Rationalize the denominator in the first term.

(sqrt2sqrt3)/(sqrt3sqrt3)+sqrt3/sqrt2+(2sqrt1)/sqrt24

Apply rule sqrtasqrta=a.

(sqrt2sqrt3)/3+sqrt3/sqrt2+(2sqrt1)/sqrt24

Apply rule sqrtasqrtb=sqrt(ab).

sqrt6/3+sqrt3/sqrt2+(2sqrt1)/sqrt24

Rationalize the denominator in the second term.

sqrt6/3+(sqrt3sqrt2)/(sqrt2sqrt2)+(2sqrt1)/sqrt24

Apply rule sqrtasqrta=a.

sqrt6/3+(sqrt3sqrt2)/2+(2sqrt1)/sqrt24

Apply rule sqrtasqrtb=sqrtab.

sqrt6/3+sqrt6/2+(2sqrt1)/sqrt24

Rationalize the denominator in the last term.

sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/(sqrt24sqrt24)

Apply rule sqrtasqrta=a.

sqrt6/3+sqrt6/2+(2sqrt1sqrt24)/24

Apply rule sqrtasqrtb=sqrt(ab).

sqrt6/3+sqrt6/2+(2sqrt24)/24

Prime factorize sqrt24.

sqrt6/3+sqrt6/2+(2sqrt(2^2xx2xx3))/24

Apply rule sqrt(a^2)=a.

sqrt6/3+sqrt6/2+(2xx2sqrt6)/24

Simplify.

sqrt6/3+sqrt6/2+(4sqrt6)/24

Simplify (4sqrt6)/24 to sqrt6/6.

sqrt6/3+sqrt6/2+sqrt6/6

The least common denominator (LCD) is 6.

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

sqrt6/3xxcolor(magenta)2/color(magenta)2+sqrt6/2xxcolor(teal)3/color(teal)3+sqrt6/6

Simplify.

(2sqrt6)/6+(3sqrt6)/6+sqrt6/6

Combine numerators.

((2sqrt6+3sqrt6+sqrt6))/6

Simplify.

(6sqrt6)/6

Simplify.

sqrt6