Dividing Radicals with Variables Worksheet
After students learn to simplify radicals and multiply radical expressions, the next skill is dividing radicals with variables. This topic is common in grade 8 math and Algebra 1 because it builds fluency with square roots, exponents, and simplifying expressions.
This free printable dividing radicals with variables worksheet gives students focused practice dividing square roots, combining expressions under one radical, and rewriting answers in simplest radical form. The directions assume variables represent positive real numbers.
What does it mean to divide radicals with variables?
A helpful rule for dividing radicals is:
$$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$$
In practice, students often rewrite a quotient of radicals as a single radical, simplify what is inside, and then pull out any perfect square factors (including perfect square variable powers). For example:
$$\frac{\sqrt{72x^2}}{\sqrt{8x}}=\sqrt{\frac{72x^2}{8x}}=\sqrt{9x}=3\sqrt{x}$$
The problems on this worksheet are written to reinforce that βcombine first, simplify afterβ routine.
What students practice on this worksheet
This worksheet includes 16 problems that build fluency with dividing radicals that include variables. Students practice:
- simplifying variable powers when dividing (like \( \sqrt{x^6}\div\sqrt{x^2} \))
- reducing coefficients inside radicals (like \( \sqrt{100x^4}\div\sqrt{25x^2} \))
- quotients that simplify to a whole expression (like \( \sqrt{48x^3}\div\sqrt{3x} \))
- answers that still include a radical (like \( \sqrt{16x^2}\div\sqrt{4x} \))
The clean two-column layout makes this worksheet easy to use for independent practice, homework, review days, or test preparation.
How to use this worksheet
- Classwork: assign the first column, then finish the second for homework
- Homework: assign all 16 for full practice
- Quick check: assign only the odd-numbered problems
- Targeted support: focus on problems that leave a radical in the final answer
An answer key is included for quick checking. If students are skipping simplification steps, they usually need more practice factoring out perfect squares or canceling variable powers correctly.
Extend the skill
Once students are comfortable dividing radicals with variables, they are ready for mixed practice and for rewriting fractions so there are no radicals in the denominator. Good next steps include:
- mixed radical operations
- rationalizing the denominator
