Explore: Multiplying Radicals Solutions

Multiplying Radicals Solutions

When radicals are multiplied: $a \sqrt{b} \cdot c \sqrt{d} = a c \sqrt{b d}$
 The coefficients are multiplied and remain outside of the radical symbol. 
The radicands are multiplied and then simplified further if possible.

Example 4

Simplify.

$7 \sqrt[3]{6 p^{5} q^{3}} \cdot 4 \sqrt[3]{12 p^{11} q^{5}}$

$28 \sqrt[3]{72 p^{16} q^{8}} 72 = 2^{3} \cdot 3^{2}$

$28 (2^{\frac{3}{3}} 3^{\frac{2}{3}} p^{\frac{16}{3}} q^{\frac{8}{3}}) 28 ({2^{1}}^{} 3^{\frac{2}{3}} p^{5 \frac{1}{3}} q^{2 \frac{2}{3}}) 28 (2 p^{5} q^{2}) \sqrt[3]{3^{2} p q^{2}} 56 p^{5} q^{2} \sqrt[3]{9 p q^{2}}$

Explain

Multiply coefficients 
Product Rule (Add exponents with like bases) 
Prime factor of the numerical radicand 
Radicand with rational exponents 
Simplify

Note

For simplicity, only multiply together radicals with the same index. If the index is different, you need to find the LCD for all rational exponents because fraction rules also apply to exponents.

Example 5

Simplify.

$- 2 x \sqrt{10 x y z} \cdot \sqrt{10 x^{3} y z}$

$- 2 x \sqrt{10^{2} x^{4} y^{2} z^{2}} - 2 x (10^{\frac{2}{2}} x^{\frac{4}{2}} y^{\frac{2}{2}} z^{\frac{2}{2}}) - 2 x (10^{1} x^{2} y^{1} z^{1}) - 20 | x^{3} y z |$