How do you multiply #sqrt(2)(sqrt 8 + sqrt 4)#?
2 Answers
Explanation:
Method 1: Multiply first then simplify
The distributive property (of multiplication over addition) tells us that
Further as a property of exponents (and therefore of square roots)
So
Method 2: Simplify roots then multiply
Explanation:
#sqrt(2)(sqrt 8 + sqrt 4)=sqrt(2)*sqrt(8)+sqrt(2)*sqrt(4)# for the distributive property.#sqrt(2)*sqrt(8)+sqrt(2)*sqrt(4)=sqrt(2*8)+sqrt(2*4)=sqrt(2^4)+sqrt(2^3)# because we can write the product of two square roots as the square roots of the product. I rewrote the products in exponential form, it is easier.- Now, if you remember, the square root of x means x at the exponent of 1/2:
#sqrt(x)=x^(1/2)# .
So in this case we can write:
#sqrt(2^4)+sqrt(2^3)=2^(4/2)+2^(3/2)# - Now we can solve the equation:
#2^(4/2)+2^(3/2)=2^2+2^(1+1/2)=4+2*2^(1/2)=4+2sqrt(2)=2(2+sqrt(2))# .