Factoring expression w/and w/o fractional exponents..?

Question

1. x^(-3/2) + 2x^(-1/2) + x^(1/2)

2. (x^2 +1)^(1/2) + 2(x^2 +1)^(-1/2)

3. (a^2 +1)^2 - 7(a^2 +1) +10

4. 3x^2( 4x-12)^2 + x^3(2)(4x-12)(4)

5. 3(2x-1)^2 (2) (x+3)^(1/2) + (2x-1)^3 (1/2) (x+3)^(-1/2)

please show the work so i understand the methods used

much appreciated

Answer 1

1. -3/2 is the smallest exponent so we factor out x^(-3/2)

x^(-3/2)(1+2x+x^2) 2x because when multiplying things with the same base you add powers, and same with x^2.

2.(-1/2) is the smallest here:

(x^2+1)^(-1/2)((x^2+1)+2)

(x^2+1)^(-1/2)(x^2+3)

3. 10 doesnt have an (a^2+1) term, but we'll factor out of the first two:

(a^2+1)((a^2+1)-7) + 10

(a^2+1)(a^2-6) + 10

4. We have x^2 and (4x-12) in common.

(x^2)(4x-12)(3(4x-12)+8x)

5. Take out (2x-1)^2 and (x+3)^(-1/2)

((2x-1)^2)((x+3)^(-1/2))(6(x+3)+(2x-1)...

((2x-1)^2)((x+3)^(-1/2))(8x+17)

Basically you have to find what's in common. If there are fraction or negative exponents, take out the one with the smallest exponent. Then you'll have to add the opposite of what you took out to the terms. For instance:

If you have ax^(1/2) + bx^(-3/2)

You take out x^(-3/2) because -3/2 is the smallest exponent. The opposite is 3/2 so you add that to the exponents to get:

x^(-3/2)(ax^((1/2)+(3/2))+bx^((-3/2)+(...

=

x^(-3/2)(ax^2+bx)

Ask if you have questions.