9.4: Multiplication of Square Root Expressions
The Product Property of Square Roots
In our work with simplifying square root expressions, we noted that
\(\sqrt{xy} = \sqrt{x} \sqrt{y}\)
Since this is an equation, we may write it as:
\(\sqrt{x} \sqrt{y} = \sqrt{xy}\)
To multiply two square root expressions, we use the product property of square roots.
\(\sqrt{x} \sqrt{y} = \sqrt{xy}\)
The product of square roots is the square root of the product
In practice, it is usually easier to simplify the square root expressions before actually performing the multiplication. To see this, consider the following product:
\(\sqrt{8} \sqrt{48}\)
We can multiply these square roots in either of two ways:
\(\sqrt{4 \cdot 2} \sqrt{16 \cdot 3} = (2 \sqrt{2})(4 \sqrt{3}) = 2 \cdot 4 \sqrt{2 \cdot 3} = 8 \sqrt{6}\)
\(\sqrt{8} \sqrt{48} = \sqrt{8 \cdot 48} = \sqrt{384} = \sqrt{64 \cdot 6} = 8 \sqrt{6}\)
Notice that in the second method, the expanded term (the third expression, \(\sqrt{384}\)) may be difficult to factor into a perfect square and some other number.
Multiplication Rule for Square Root Expressions
The preceding example suggests that the following rule for multiplying two square root expressions.
- Simplify each square root expression, if necessary.
- Perform the multiplication.
- Simplify, if necessary.
Sample Set A
Find each of the following products.
\(\sqrt{3} \sqrt{6} = \sqrt{3 \cdot 6} = \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)
\(\sqrt{8} \sqrt{2} = 2 \sqrt{2} \sqrt{2} = 2 \sqrt{2 \cdot 2} = 2 \sqrt{4} = 2 \cdot 2 = 4\)
This product might be easier if we were to multiply first then simplify.
\(\sqrt{8} \sqrt{2} = \sqrt{8 \cdot 2} = \sqrt{16} = 4\)
\(\sqrt{20} \sqrt{7} = \sqrt{4} \sqrt{5} \sqrt{7} = 2 \sqrt{5 \cdot 7} = 2 \sqrt{35}\)
\(\begin{array}{flushleft}
\sqrt{5a^3} \sqrt{27a^5} = (a \sqrt{5a})(3a^2 \sqrt{3a}) &= 3a^3 \sqrt{15a^2}\\
&= 3a^3 \cdot a \sqrt{15}\\
&= 3a^4 \sqrt{15}
\end{array}\)
\(\begin{array}{flushleft}
\sqrt{(x+2)^7} \sqrt{x-1} = \sqrt{(x+2)^6(x+2)} \sqrt{x-1} &= (x+2)^3 \sqrt{(x+2)} \sqrt{x-1}\\
&= (x+2)^3 \sqrt{(x+2)(x-1)}\\
\text{or} &= (x+2)^3 \sqrt{x^2 + x - 2}
\end{array}\)
\(\begin{array}{flushleft}
\sqrt{3}(7+\sqrt{6})=7 \sqrt{3}+\sqrt{3} \sqrt{6} &=7 \sqrt{3}+\sqrt{18} \\
&=7 \sqrt{3}+\sqrt{9 \cdot 2} \\
&=7 \sqrt{3}+3 \sqrt{2}
\end{array}\)
\(\begin{array}{flushleft}
\sqrt{6}(\sqrt{2} - \sqrt{10}) &= \sqrt{6} \sqrt{2} - \sqrt{6}\sqrt{10}\\
&= \sqrt{12} - \sqrt{60}\\
&= \sqrt{4 \cdot 3} - \sqrt{4 \cdot 15}\\
&= 2 \sqrt{3} - 2 \sqrt{15}
\end{array}\)
\(\begin{array}{flushleft}
\sqrt{45 a^{6} b^{3}}\left[\sqrt{9 a b}-\sqrt{5(b-3)^{3}}\right] &=3 a^{3} b \sqrt{5 b}[3 \sqrt{a b}-(b-3) \sqrt{5(b-3)}] \\
&=9 a^{3} b \sqrt{5 a b^{2}}-3 a^{3} b(b-3) \sqrt{25 b(b-3)} \\
&=9 a^{3} b^{2} \sqrt{5 a}-3 a^{3} b(b-3) \cdot 5 \sqrt{b(b-3)} \\
&=9 a^{3} b^{2} \sqrt{5 a}-15 a^{3} b(b-3) \sqrt{b(b-3)}
\end{array}\)
Practice Set A
Find each of the following products.
\(\sqrt{5} \sqrt{6}\)
- Answer
-
\(\sqrt{30}\)
\(\sqrt{32} \sqrt{2}\)
- Answer
-
\(8\)
\(\sqrt{x+4} \sqrt{x+3}\)
- Answer
-
\(\sqrt{(x+4)(x+3)}\)
\(\sqrt{8m^5n} \sqrt{20m^2n}\)
- Answer
-
\(4m^3n \sqrt{10m}\)
\(\sqrt{9(k-6)^3} \sqrt{k^2 - 12k + 36}\)
- Answer
-
\(3(k-6)^2 \sqrt{k-6}\)
\(\sqrt{3} \sqrt{2} + \sqrt{5}\)
- Answer
-
\(\sqrt{6} + \sqrt{15}\)
\(\sqrt{2a} (\sqrt{5a} - \sqrt{8a^3})\)
- Answer
-
\(a \sqrt{10} - 4a^2\)
\(\sqrt{32m^5n^8} \sqrt{2mn^2} - \sqrt{n^7}\)
- Answer
-
\(8m^3n^2 \sqrt{n} - 8m^2n^5\sqrt{5m}\)
Exercises
\(\sqrt{2} \sqrt{10}\)
- Answer
-
\(2 \sqrt{5}\)
\(\sqrt{3} \sqrt{15}\)
\(\sqrt{7} \sqrt{8}\)
- Answer
-
\(2 \sqrt{14}\)
\(\sqrt{20} \sqrt{3}\)
\(\sqrt{32} \sqrt{27}\)
- Answer
-
\(12 \sqrt{6}\)
\(\sqrt{45} \sqrt{50}\)
\(\sqrt{5} \sqrt{5}\)
- Answer
-
\(5\)
\(\sqrt{7} \sqrt{7}\)
\(\sqrt{8} \sqrt{8}\)
- Answer
-
\(8\)
\(\sqrt{15} \sqrt{15}\)
\(\sqrt{48} \sqrt{27}\)
- Answer
-
\(36\)
\(\sqrt{80} \sqrt{20}\)
\(\sqrt{5} \sqrt{m}\)
- Answer
-
\(\sqrt{5m}\)
\(\sqrt{7} \sqrt{a}\)
\(\sqrt{6} \sqrt{m}\)
- Answer
-
\(\sqrt{6m}\)
\(\sqrt{10} \sqrt{h}\)
\(\sqrt{20} \sqrt{a}\)
- Answer
-
\(2 \sqrt{5a}\)
\(\sqrt{48} \sqrt{x}\)
\(\sqrt{75} \sqrt{y}\)
- Answer
-
\(5 \sqrt{3y}\)
\(\sqrt{200} \sqrt{m}\)
\(\sqrt{a} \sqrt{a}\)
- Answer
-
\(a\)
\(\sqrt{x} \sqrt{x}\)
\(\sqrt{y} \sqrt{y}\)
- Answer
-
\(y\)
\(\sqrt{h} \sqrt{h}\)
\(\sqrt{3} \sqrt{3}\)
- Answer
-
\(3\)
\(\sqrt{6} \sqrt{6}\)
\(\sqrt{k} \sqrt{k}\)
- Answer
-
\(k\)
\(\sqrt{m} \sqrt{m}\)
\(\sqrt{m^2} \sqrt{m}\)
- Answer
-
\(m \sqrt{m}\)
\(\sqrt{a^2} \sqrt{a}\)
\(\sqrt{x^3} \sqrt{x}\)
- Answer
-
\(x^2\)
\(\sqrt{y^3} \sqrt{y}\)
\(\sqrt{y} \sqrt{y^4}\)
- Answer
-
\(y^2 \sqrt{y}\)
\(\sqrt{k} \sqrt{k^6}\)
\(\sqrt{a^3} \sqrt{a^5}\)
- Answer
-
\(a^4\)
\(\sqrt{x^3} \sqrt{x^7}\)
\(\sqrt{x^9} \sqrt{x^3}\)
- Answer
-
\(x^6\)
\(\sqrt{y^3} \sqrt{y^4}\)
- Answer
-
\(y^3 \sqrt{y}\)
\(\sqrt{x^8} \sqrt{x^5}\)
\(\sqrt{x+2} \sqrt{x-3}\)
- Answer
-
\(\sqrt{(x+2)(x-3)}\)
\(\sqrt{a-6} \sqrt{a+1}\)
\(\sqrt{y+3} \sqrt{y - 2}\)
- Answer
-
\(\sqrt{(y+3)(y-2)}\)
\(\sqrt{h+1} \sqrt{h-1}\)
\(\sqrt{x+9} \sqrt{(x+9)^2}\)
- Answer
-
\((x+9) \sqrt{x+9}\)
\(\sqrt{y-3} \sqrt{(y-3)^5}\)
\(\sqrt{3a^2} \sqrt{15a^3}\)
- Answer
-
\(3a^2 \sqrt{5a}\)
\(\sqrt{2m^4n^3} \sqrt{14m^5n}\)
\(\sqrt{12(p-q)^3} \sqrt{3(p-q)^5}\)
- Answer
-
\(6(p-q)^4\)
\(\sqrt{15a^2(b+4)^4} \sqrt{21a^3(b+4)^5}\)
\(\sqrt{125m^5n^4r^8} \sqrt{8m^6r}\)
- Answer
-
\(10m^5n^2r^4 \sqrt{10mr}\)
\(\sqrt{7(2k-1)^{11}(k+1)^3} \sqrt{14(2k-1)^{10}}\)
\(\sqrt{y^3} \sqrt{y^5} \sqrt{y^2}\)
- Answer
-
\(y^5\)
\(\sqrt{x^6} \sqrt{x^2} \sqrt{x^9}\)
\(\sqrt{2a^4} \sqrt{5a^3} \sqrt{2a^7}\)
- Answer
-
\(2a^7 \sqrt{5}\)
\(\sqrt{x^n} \sqrt{x^n}\)
\(\sqrt{y^2n} \sqrt{y^4n}\)
- Answer
-
\(y^{3n}\)
\(\sqrt{a^{2n + 5}} \sqrt{a^3}\)
\(\sqrt{2m^{3n + 1}} \sqrt{10m^{n + 3}}\)
- Answer
-
\(2m^{2n + 2} \sqrt{5}\)
\(\sqrt{75(a-2)^7} \sqrt{48a - 96}\)
\(\sqrt{2} (\sqrt{8} + \sqrt{6})\)
- Answer
-
\(2(2 + \sqrt{3})\)
\(\sqrt{5}(\sqrt{3} + \sqrt{7})\)
\(\sqrt{3} (\sqrt{x} + \sqrt{2})\)
- Answer
-
\(\sqrt{3x} + \sqrt{6}\)
\(\sqrt{11}(\sqrt{y} + \sqrt{3})\)
\(\sqrt{8}(\sqrt{a} - \sqrt{3a})\)
- Answer
-
\(2\sqrt{2a} - 2\sqrt{6a}\)
\(\sqrt{x}(\sqrt{x^3} - \sqrt{2x^4})\)
\(\sqrt{y}(\sqrt{y^5} \sqrt{3y^3})\)
- Answer
-
\(9y^2(y + \sqrt{3})\)
\(\sqrt{8a^5}(\sqrt{2a} - \sqrt{6a^{11}})\)
\(\sqrt{12m^3} (\sqrt{6m^7} - \sqrt{3m})\)
- Answer
-
\(6m^2 (m^3 \sqrt{2} - 1)\)
\(\sqrt{5x^4y^3} (\sqrt{8xy} - 5\sqrt{7x})\)
Exercises for Review
Factor \(a^4y^4 - 25w^2\)
- Answer
-
\((a^2y^2 + 5w)(a^2y^2 - 5w)\)
Find the slope of the line that passes through the points \((-5, 4)\) and \((-3, 4)\)
Perform the indicated operations:
\(\dfrac{15 x^{2}-20 x}{6 x^{2}+x-12} \cdot \dfrac{8 x+12}{x^{2}-2 x-15} \div \dfrac{5 x^{2}+15 x}{x^{2}-25}\)
- Answer
-
\(\dfrac{4(x+5)}{(x+3)^2}\)
Simplify \(\sqrt{x^4y^2z^6}\) by removing the radical sign
Simplify \(\sqrt{12x^3y^5z^8}\)
- Answer
-
\(2xy^2z^4 \sqrt{3xy}\)