How to Multiply Radical Expressions

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Multiplying Radical Expressions

To multiply two square roots, radicands should be multiplied together and then write the result under a single radical. Sometimes it could be necessary to further simplify the radical expression. This procedure always stays the same, regardless of the index's value.

Steps of Multiplying Radical Expressions

First, the numbers and expressions outside of the radicals should be multiplied.
Then, the numbers and expressions inside of the radicals should be multiplied.
If necessary, simplify.

Example1

Find the answer: \(3 \ \sqrt{7} \times 5 \ \sqrt{14}\)

Solution:

The numbers and expressions outside and inside of the radicals should be multiplied:
\(3 \ \sqrt{7} \times 5 \ \sqrt{14} \ = \ 3 \times 5 \ \sqrt{7 \times 14} \ = \ 15 \ \sqrt{98}\)
Simplifying:
\(15 \ \sqrt{98} \ = \ 15 \times 7 \ \sqrt{2} \ = \ 105 \ \sqrt{2}\)

Example2

Find the answer: \(2 \ \sqrt[3]{30} \times 3 \ \sqrt[3]{12}\)

Solution:

The numbers and expressions outside and inside of the radicals should be multiplied:
\(2 \ \sqrt[3]{30} \times 3 \ \sqrt[3]{12} \ = \ 2 \times 3 \ \sqrt[3]{30 \times 12} \ = \ 6 \ \sqrt[3]{360}\)
Simplifying:
\(6 \ \sqrt[3]{360} \ = \ 6 \ \sqrt[3]{45 \times 8} \ = \ 6 \times 2 \ \sqrt[3]{45} \ = \ 12 \ \sqrt[3]{45}\)

Multiplying Radical Expressions

Think of this lesson as more than a rule to memorize. Multiplying Radical Expressions is about roots, simplification, restrictions, and extraneous solutions. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Radicals undo powers. For square roots, \(\sqrt{a}\) means the nonnegative number whose square is \(a\), and perfect-square factors help simplify expressions.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Look for perfect-square factors.
Simplify the radical before combining terms.
When solving, isolate the radical before squaring.
Check for extraneous solutions.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Multiplying Radical Expressions

1) Find the product: \(\sqrt{3} \times \sqrt{12}\)

2) Find the product: \(\sqrt{5} \times \sqrt{20}\)

3) Find the product: \(\sqrt{6} \times \sqrt{15}\)

4) Find the product: \((2\sqrt{7})(3\sqrt{14})\)

5) Find the product, assuming \(x\) is nonnegative: \(\sqrt{8x} \times \sqrt{18x}\)

6) Find the product, assuming \(a\) is nonnegative: \(\sqrt{5a^2} \times \sqrt{20a}\)

7) Find the product: \((3\sqrt{2})(4\sqrt{18})\)

8) Find the product, assuming \(m\) is nonnegative: \(\sqrt{12m^3} \times \sqrt{3m}\)

9) Find the product, assuming \(x\) is nonnegative: \(\sqrt{7x^5} \times \sqrt{28x^3}\)

10) Find the product, assuming \(y\) is nonnegative: \((2\sqrt{3y})(5\sqrt{27y^3})\)

11) Find the product, assuming variables are nonnegative: \(\sqrt{10a^3b} \times \sqrt{40ab^5}\)

12) Multiply: \((\sqrt{6} \ + \ \sqrt{2})(\sqrt{6} \ - \ \sqrt{2})\)

13) Multiply: \((3\sqrt{5} \ + \ 2)(3\sqrt{5} \ - \ 2)\)

14) Find the product, assuming variables are nonnegative: \(\sqrt{18p^4q} \times \sqrt{50p^2q^3}\)

15) Multiply: \((\sqrt{x} \ + \ 4)(\sqrt{x} \ - \ 4)\)

16) Multiply: \((2\sqrt{3} \ + \ \sqrt{5})^2\)

17) Find the product, assuming \(a\) is nonnegative: \(\sqrt{24a^5} \times \sqrt{54a^3}\)

18) Multiply: \((\sqrt{7} \ + \ \sqrt{3})(\sqrt{7} \ - \ 2\sqrt{3})\)

19) Multiply: \((5\sqrt{2} \ - \ \sqrt{8})(3\sqrt{2} \ + \ \sqrt{18})\)

20) Find the product, assuming variables are nonnegative: \(\sqrt{45x^7y^2} \times \sqrt{80x^5y^4}\)

Answers

1)Find the product: \(\sqrt{3} \times \sqrt{12}\)