Algebra

sqrt(9xy) / sqrt(3x^2y)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Write as one fraction under radical

When you divide two square roots, you can combine them into a single square root of a fraction. This makes it easier to simplify.
2

Simplify the fraction inside the radical

Separate the numbers and each variable into their own fractions. This lets us reduce each part one at a time.
3

Reduce each part of the fraction

9 divided by 3 is 3. x divided by x² is 1/x. y divided by y is 1. So the inside simplifies to 3/x.
4

Rationalize the denominator

We don't leave a square root in the denominator. Multiply top and bottom by √x to move it out of the denominator. This is called rationalizing.

Understanding this problem

Learning Insight

The rule √a / √b = √(a/b) works because squaring both sides gives a/b on both sides — they are truly equal. Simplifying under one radical first is almost always faster than working with two separate radicals.

Quick Tip

Spot matching variables or numbers in the numerator and denominator right away — they cancel before you even take a square root, cutting your work in half.

Common Mistake

Students often cancel the y terms but forget to reduce x/x² correctly, writing x instead of 1/x. Remember x/x² = 1/x, not x.