## How to Multiply 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:
$$6 \ \sqrt[3]{360} \ = \ 6 \ \sqrt[3]{45 \times 8} \ = \ 6 \times 2 \ \sqrt[3]{45} \ = \ 12 \ \sqrt[3]{45}$$

### 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: $$\sqrt{17 \ t^5 \ r^6} \times \sqrt{51 \ t^3 \ r^2} \ =$$

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

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