Multiplying Radicals
In math, you can multiply just about everything, including radicals!
Let's review what a radical is: a radical is a root. If there's no superscript number in front of the radical, it's a square root (the 2 in front of the radical is assumed if there is no other number), which asks "what number, times itself, equals this number?"
Because a radical is a square root, a radical squared is the number under the radical:
$\sqrt{12}^2=(\sqrt{12})(\sqrt{12})=12$
When radicals are multiplied by other radicals, you can multiply the numbers and keep the product under the radical. If you have a hard time remembering this rule, you can always test the rule with square roots that you know.
$(\sqrt{4})(\sqrt{9})=\sqrt{36}=6$
Can you prove that that works?
$(\sqrt{4})(\sqrt{9})=2 \times 3 =6$
The answers are the same. This technique works.
When radicals are preceded by a coefficient, you multiply the radicals and the coefficients. To remember this, think about the radicals as variables!
First, remember the rules for multiplying with variables, you multiply the number AND the variable:
$(5x)(8y)=40xy$
$(2x)(4x)=8x^2$
The same rules and patterns apply to radicals:
$(2\sqrt{7})(3\sqrt{3})=6\sqrt{21}$
$(2\sqrt{13})(2\sqrt{2})=4\sqrt{26}$
$(5\sqrt{7})(2\sqrt{7})=10\sqrt{49}=10 \times 7=70$
After you multiply radicals, you may need to Simplify Radicals. Check out that lesson for a refresher.
Practice Problems:
Multiplying Radicals
Find the products. Simplify answers.
- $\sqrt{2}\times \sqrt{20}=$
- $\sqrt{8}\times \sqrt{3}=$
- $\sqrt{35}\times \sqrt{5}=$
- $\sqrt{4}\times \sqrt{56}=$
- $\sqrt{64}\times \sqrt{8}=$
- $\sqrt{18}\times \sqrt{6}=$
- $\sqrt{49}\times \sqrt{4}=$
- $\sqrt{5}\times \sqrt{50}=$
- $\sqrt{9}\times \sqrt{28}=$
- $\sqrt{63}\times \sqrt{7}=$
- $\sqrt{5}(2+\sqrt{3})=$
- $\sqrt{2}(4\sqrt{2}-7)=$
- $\sqrt{3}(\sqrt{6}+1)=$
- $\sqrt{6}(2+3\sqrt{3})=$
- $\sqrt{7}(7-\sqrt{2})=$
- $\sqrt{2}(10-10\sqrt{3})=$
- $\sqrt{5}(6\sqrt{6}+\sqrt{2})=$
- $\sqrt{7}(\sqrt{7}-2\sqrt{5})=$
- $\sqrt{2}(4\sqrt{2}-\sqrt{3})=$
- $\sqrt{3}(\sqrt{6}+2\sqrt{2})=$
Answer Key: