Algebra

(3√(6a^6)) / (3√(2a^5b))

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Rewrite as one cube root fraction

We start by recognizing both the numerator and denominator are cube roots. We can combine them into a single cube root of a fraction using the rule .
2

Combine into one cube root

We apply the quotient rule for radicals to merge the two cube roots into one. This makes it easier to simplify the expression inside.
3

Simplify the fraction inside

Divide the coefficients: . Divide the powers of : . The stays in the denominator. This gives us .
4

Rationalize the denominator

We multiply top and bottom by so the denominator becomes a perfect cube: . This removes the cube root from the denominator.
5

Simplify the denominator to b

Since , the denominator simplifies cleanly to just . The final simplified expression has a rational denominator.

Understanding this problem

Learning Insight

The quotient rule for radicals says . This works because radicals are really fractional exponents, and dividing same-base exponents means subtraction. Rationalizing the denominator is needed because we don't leave roots in the bottom of a fraction in simplified form.

Quick Tip

To rationalize a cube root denominator , always multiply by — you need the exponent inside to reach 3 (a perfect cube). Think: 'what power do I need to add to get to 3?'

Common Mistake

Students often multiply by instead of when rationalizing, giving in the denominator instead of a whole number .