How to simplify these expressions: a) (2√8 + 5√12)² ; b) (-√7+ 2√7)(√7- 2√7)  c) (2√3-√5)(2√3+√5)

1 Answer | Add Yours

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

A square root is simplified if (1) there are no perfect square factors in the radicand (2) There are no fractions in the radicand (3) there are no roots in the denominator.

(a) We begin by simplifying the roots:

`(2sqrt(8)+5sqrt(12))^2=(2sqrt(4*2)+5sqrt(4*3))^2`

`=(4sqrt(2)+10sqrt(3))^2`

Now we can expand the binomial: `(a+b)^2=a^2+2ab+b^2`

`(4sqrt(2)+10sqrt(3))^2=(4sqrt(2))^2+2(4sqrt(2))(10sqrt(3))+(10sqrt(3))^2`

`=16*2+80sqrt(6)+100*3`

`=332+80sqrt(6)`

So the simplified form is `332+80sqrt(6)`

(b) `(-sqrt(7)+2sqrt(7))(sqrt(7)-2sqrt(7))`

First we can add like terms:

`-sqrt(7)+2sqrt(7)=sqrt(7)`  and `sqrt(7)-2sqrt(7)=-sqrt(7)`

So we have:

`(-sqrt(7)+2sqrt(7))(sqrt(7)-2sqrt(7))=(sqrt(7))(-sqrt(7))=-7`

So the simplified form is -7.

(c) `(2sqrt(3)-sqrt(5))(2sqrt(3)+sqrt(5))` ` `

` ` Here we can use the difference of two squares: `(a+b)(a-b)=a^2-b^2`

So `(2sqrt(3)-sqrt(5))(2sqrt(3)+sqrt(5))=(2sqrt(3))^2-(sqrt(5))^2`

`=(4*3)-5`

`=7`

So the answer is 7.

Sources:

We’ve answered 318,928 questions. We can answer yours, too.

Ask a question