How to Multiply Radical Expressions

How to Multiply Radical Expressions

 Read,4 minutes

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:
    \(6 \ \sqrt[3]{360} \ = \ 6 \ \sqrt[3]{45 \times 8} \ = \ 6 \times 2 \ \sqrt[3]{45} \ = \ 12 \ \sqrt[3]{45}\)

Free printable Worksheets

Exercises for Multiplying Radical Expressions

1) Find the answer: \(\sqrt{5 \ n^3} \times \sqrt{5 \ n} \ =\)

2) Find the answer: \(\sqrt{35 \ p^7} \times \sqrt{14 \ p^3} \ =\)

3) Find the answer: \(\sqrt{48 \ m^{10}} \times \sqrt{9 \ m^5} \ =\)

4) Find the answer: \(\sqrt{30 \ r^9} \times \sqrt{24 \ r^2} \ =\)

5) Find the answer: \(\sqrt{14 \ m^{11}} \times \sqrt{7 \ m^7} \ =\)

6) Find the answer: \(\sqrt{18 \ x^{18}} \times \sqrt{27 \ x^{11}} \ =\)

7) Find the answer: \(\sqrt{12 \ z^{16}} \times \sqrt{8 \ z^6} \ =\)

8) Find the answer: \(\sqrt{21 \ x^{10} \ y^2} \times \sqrt{63 \ x^3 \ y^7} \ =\)

9) Find the answer: \(\sqrt{66 \ q^9 \ p^7} \times \sqrt{22 \ q^7 \ p^8} \ =\)

10) Find the answer: \(\sqrt{17 \ t^5 \ r^6} \times \sqrt{51 \ t^3 \ r^2} \ =\)

 

1) Find the answer: \(\sqrt{5 \ n^3} \times \sqrt{5 \ n} \ =\)

\(\color{red}{\sqrt{5 \ n^3} \times \sqrt{5 \ n} \ = \ \sqrt{5^2 \ n^4} \ = \ 5 \ n^2}\)

2) Find the answer: \(\sqrt{35 \ p^7} \times \sqrt{14 \ p^3} \ =\)

\(\color{red}{\sqrt{35 \ p^7} \times \sqrt{14 \ p^3} \ = \ \sqrt{5 \times 7 \ p^7} \times \sqrt{2 \times 7 \ p^3}}\) \(\color{red}{ \ = \ \sqrt{5 \times 7^2 \times 2 \ p^{2 \times 5}} \ = \ 7 \ p^5 \ \sqrt{5 \times 2} \ = \ 7 \ p^5 \ \sqrt{10}}\)

3) Find the answer: \(\sqrt{48 \ m^{10}} \times \sqrt{9 \ m^5} \ =\)

\(\color{red}{\sqrt{48 \ m^{10}} \times \sqrt{9 \ m^5} \ = \ \sqrt{3 \times 4^2 \ m^{10}} \times \sqrt{3^2 \times m^5} \ = \ \sqrt{3^3 \times 4^2 \ m^{2 \times 7} \times m}}\) \(\color{red}{ = \ 3 \times 4 \ m^7 \ \sqrt{3 \times m} \ = 12 \ m^7 \ \sqrt{3 \ m}}\)

4) Find the answer: \(\sqrt{30 \ r^9} \times \sqrt{24 \ r^2} \ =\)

\(\color{red}{\sqrt{30 \ r^9} \times \sqrt{24 \ r^2} \ = \ \sqrt{5 \times 3 \times 2 \ r^9} \times \sqrt{2^3 \times 3 \ r^2} \ = \ \sqrt{5 \times 3^2 \times 2^4 \ r^{2 \times 5} \times r}}\) \(\color{red}{ \ = \ 7 \ p^5 \ \sqrt{5 \times r} = \ 3 \times 2^2 \ r^5 \ \sqrt{10} \ = \ 12 \ r^5 \ \sqrt{10}}\)

5) Find the answer: \(\sqrt{14 \ m^{11}} \times \sqrt{7 \ m^7} \ =\)

\(\color{red}{\sqrt{14 \ m^{11}} \times \sqrt{7 \ m^7} \ = \ \sqrt{2 \times 7^2 \ m^{18}} \ = \ 7 \ m^9 \ \sqrt{2}}\)

6) Find the answer: \(\sqrt{18 \ x^{18}} \times \sqrt{27 \ x^{11}} \ =\)

\(\color{red}{\sqrt{18 \ x^{18}} \times \sqrt{27 \ x^{11}} \ = \ \sqrt{2 \times 3^2 \ x^{18}} \times \sqrt{3^3 \ x^{11}} \ = \ \sqrt{2 \times 3^5 \ x^{28} \times x}}\) \(\color{red}{ \ = \ 3^2 \ x^{14} \ \sqrt{2 \times 3 \times x} = \ 9 \ x^{14} \ \sqrt{6x}}\)

7) Find the answer: \(\sqrt{12 \ z^{16}} \times \sqrt{8 \ z^6} \ =\)

\(\color{red}{\sqrt{12 \ z^{16}} \times \sqrt{8 \ z^6} \ = \ \sqrt{3 \times 2^5 \ z^{22}} \ = \ 4 \ z^{11} \ \sqrt{6}}\)

8) Find the answer: \(\sqrt{21 \ x^{10} \ y^2} \times \sqrt{63 \ x^3 \ y^7} \ =\)

\(\color{red}{\sqrt{21 \ x^{10} \ y^2} \times \sqrt{63 \ x^3 \ y^7} \ = \ \sqrt{3^3 \times 7^2 \ x^{13} \ y^9} \ = \ 3 \times 7 \ x^6 \ y^{4} \ \sqrt{3 \ x \ y} \ = \ 21 \ x^6 \ y^{4} \ \sqrt{3 \ x \ y}}\)

9) Find the answer: \(\sqrt{66 \ q^9 \ p^7} \times \sqrt{22 \ q^7 \ p^8} \ =\)

\(\color{red}{\sqrt{66 \ q^9 \ p^7} \times \sqrt{22 \ q^7 \ p^8} \ = \ \sqrt{2 \times 3 \times 11^2 \ q^{16} \ p^{15}} \ = \ 11 \ q^8 \ p^7 \ \sqrt{2 \times 3 \ p} \ = \ 11 \ q^8 \ p^7 \ \sqrt{6 \ p}}\)

10) Find the answer: \(\sqrt{17 \ t^5 \ r^6} \times \sqrt{51 \ t^3 \ r^2} \ =\)

\(\color{red}{\sqrt{17 \ t^5 \ r^6} \times \sqrt{51 \ t^3 \ r^2} \ = \ \sqrt{3 \times 17^2 \ t^8 \ r^8} \ = \ 17 \ t^4 \ r^4 \ \sqrt{3}}\)

Multiplying Radical Expressions Practice Quiz