I’ve Run Into An Algebra Problem And Need Help, Please.

Question

Simplify: (3√45)-(5√5)/(3√5)-(5√3)

(A) 3 - (√5/3)

(B) (-2√15)/(3) - 2

(C) - 2 + 2 (√5/3)

(D) (135 - 25 √15)/(-30)

(E) None of the above

At first, I had gotten (E) None of the above. But then I realized that the decimal form of

(B) (-2√15)/(3) - 2 is -4.5819889. Which is also the decimal form of (3√45)-(5√5)/(3√5)-(5√3). Would my answer be (E) None of the above or (B) (-2√15)/(3) - 2 ?

Answer 1

(3√45)-(5√5)/(3√5)-(5√3) this is a fraction and you want to rationalize the denominator to simplify it

(3√45)-(5√5)

(3√5)-(5√3)

Simplify the numerator

(3√45)-(5√5)

(3√9√5)-(5√5)

(3*3√5)-(5√5)

(9√5)-(5√5)

4√5

Now we have

4√5/(3√5)-(5√3)

Next we use the conjugate to rationalize the denominator

4√5 * (3√5)+(5√3)

(3√5)-(5√3)* (3√5)+(5√3)

4*3*5 + 4*5√5√3

3*3*5 + 3√5*5√3 -5√3*3√5 -5*5*3

60 + 20√15

45 +15√15 -15√15 -75

60 + 20√15

(-30)

Regroup

(20√15/-30) + 60/-30

(-2√15/3) - 2

You can plug this into a calculator to check it.

But it might look like this

(-2√15 - 6)/3

But this is the same when you regroup the terms as above and gives the same value of -4.581988897

A radical expression is said to be in its simplest form if there are no perfect square factors other than 1 in the radicand. No radicals in the denominator of a fraction. Since your expression has radicals in the numerator and denominator and B gives the same value and B has no radicals in the denominator then.

Answer 2

Simplify: (3√45)-(5√5)/(3√5)-(5√3)

Answer:

3√(9·5) - 5/3 - 5√3

3·3√5 - 5/3 - 5√3

9√5 - 5/3 - 5√3

8√5 - 5/3

Yes: None of the above.