Simplify the following expressions

Question

(1.) (1-√7)(1+√7)=

 

 

(2.) √3(2√3+√6)=

Answer 1

#1 is a special trinomial. It has the following formula:

(A+B)(A-B) = A² - B²

 

So in your problem, A=1 and B=√7

If you plug it in to A² - B², your answer is going to be: (1)² - (√7)²

This simplifies to 1-7, or -6. 

 

Another way you could solve it would be by using the FOIL method, but recognizing it as a special trinomial will save you some time!

 

 

For #2, you need to distribute the number outside of the parentheses to each number inside the parentheses:

√3(2√3) + √3(√6)

 

Then, you simplify.

 

The first part, √3(2√3), could essentially be written like this: 2 * √3 * √3.

Whenever you multiply two identical radicals, the radicals cancel each other out and you end up with: 2 * 3, or 6.

 

The second part, since you are multiplying two radicals, you just multiply the numbers and put them inside the radical:

(√3)(√6) = √(3*6), or √18

 

Then, you simplify the remaining radical by identifying any perfect squares in it. What numbers multiply to 18 and include a perfect square? 9 * 2 = 18, and 9 is a perfect square!

 

√18 is changed to √9 * √2.

 

√9 * √2 is simplified to 3 * √2, or 3√2.

 

Finally, your overall answer is  6 + 3√2.

Answer 2

(1.) (1-√7)(1+√7)=

 

FOIL it out:

 

= 1*1 + 1√7 - 1√7 - √7√7 = 1 - 7 = -6

 

 

(2.) √3(2√3+√6)=

 

= 2√3√3 + √3√6

 

= 2*3 + √18 = 6 + √(9*2) = 6 + 3√2