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Learning Objectives
By the end of this section, you will be able to:
- Use the Product Property to simplify square roots
- Use the Quotient Property to simplify square roots
In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that \(\sqrt{50}\) is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use [link].
But what if we want to estimate \(\sqrt{500}\)? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.
A square root is considered simplified if its radicand contains no perfect square factors.
Definition: SIMPLIFIED SQUARE ROOT
\(\sqrt{a}\) is considered simplified if a has no perfect square factors.
So \(\sqrt{31}\) is simplified. But \(\sqrt{32}\) is not simplified, because 16 is a perfect square factor of 32.
Use the Product Property to Simplify Square Roots
The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that \((ab)^m=a^{m}b^{m}\). The corresponding property of square roots says that \(\sqrt{ab}=\sqrt{a}·\sqrt{b}\).
Definition: PRODUCT PROPERTY OF SQUARE ROOTS
If a, b are non-negative real numbers, then \(\sqrt{ab}=\sqrt{a}·\sqrt{b}\).
We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in Example.
How To Use the Product Property to Simplify a Square Root
Example \(\PageIndex{1}\)
Simplify: \(\sqrt{50}\).
- Answer
-
Example \(\PageIndex{2}\)
Simplify: \(\sqrt{48}\).
- Answer
-
\(4\sqrt{3}\)
Example \(\PageIndex{3}\)
Simplify: \(\sqrt{45}\).
- Answer
-
\(3\sqrt{5}\)
Notice in the previous example that the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\), which is the product of an integer and a square root. We always write the integer in front of the square root.
Definition: SIMPLIFY A SQUARE ROOT USING THE PRODUCT PROPERTY.
- Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
- Use the product rule to rewrite the radical as the product of two radicals.
- Simplify the square root of the perfect square.
Example \(\PageIndex{4}\)
Simplify: \(\sqrt{500}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{500}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor}}&{\sqrt{100·5}}\\ {\text{Rewrite the radical as the product of two radicals}}&{\sqrt{100}·\sqrt{5}}\\ {\text{Simplify}}&{10\sqrt{5}}\\ \end{array}\]
Example \(\PageIndex{5}\)
Simplify: \(\sqrt{288}\).
- Answer
-
\(12\sqrt{2}\)
Example \(\PageIndex{6}\)
Simplify:\(\sqrt{432}\).
- Answer
-
\(12\sqrt{3}\)
We could use the simplified form \(10\sqrt{5}\) to estimate \(\sqrt{500}\). We know \(\sqrt{5}\) is between 2 and 3, and \(\sqrt{500}\) is \(10\sqrt{5}\). So \(\sqrt{500}\) is between 20 and 30.
The next example is much like the previous examples, but with variables.
Example \(\PageIndex{7}\)
Simplify: \(\sqrt{x^3}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{x^3}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor}}&{\sqrt{x^2·x}}\\ {\text{Rewrite the radical as the product of two radicals}}&{\sqrt{x^2}·\sqrt{x}}\\ {\text{Simplify}}&{x\sqrt{x}}\\ \end{array}\]
Example \(\PageIndex{8}\)
Simplify:\(\sqrt{b^5}\).
- Answer
-
\(b^2\sqrt{b}\)
Example \(\PageIndex{9}\)
Simplify: \(\sqrt{p^9}\).
- Answer
-
\(p^4\sqrt{p}\)
We follow the same procedure when there is a coefficient in the radical, too.
Example \(\PageIndex{10}\)
Simplify: \(\sqrt{25y^5}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{25y^5}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{25y^4·y}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{25y^4}·\sqrt{y}}\\ {\text{Simplify.}}&{5y^2\sqrt{y}}\\ \end{array}\]
Example \(\PageIndex{11}\)
Simplify: \(\sqrt{16x^7}\).
- Answer
-
\(4x^3\sqrt{x}\)
Example \(\PageIndex{12}\)
Simplify: \(\sqrt{49v^9}\).
- Answer
-
\(7v^4\sqrt{v}\)
In the next example both the constant and the variable have perfect square factors.
Example \(\PageIndex{13}\)
Simplify: \(\sqrt{72n^7}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{72n^7}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{36n^{6}·2n}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{36n^{6}}·\sqrt{2n}}\\ {\text{Simplify.}}&{6n^3\sqrt{2n}}\\ \end{array}\]
Example \(\PageIndex{14}\)
Simplify: \(\sqrt{32y^5}\).
- Answer
-
\(4y^2\sqrt{2y}\)
Example \(\PageIndex{15}\)
Simplify: \(\sqrt{75a^9}\).
- Answer
-
\(5a^4\sqrt{3a}\)
Example \(\PageIndex{16}\)
Simplify: \(\sqrt{63u^{3}v^{5}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{63u^{3}v^{5}}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{\sqrt{9u^{2}v^{4}·7uv}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\sqrt{9u^{2}v^{4}}·\sqrt{7uv}}\\ {\text{Simplify.}}&{3uv^{2}\sqrt{7uv}}\\ \end{array}\]
Example \(\PageIndex{17}\)
Simplify: \(\sqrt{98a^{7}b^{5}}\).
- Answer
-
\(7a^{3}b^{2}\sqrt{2ab}\)
Example \(\PageIndex{18}\)
Simplify: \(\sqrt{180m^{9}n^{11}}\).
- Answer
-
\(6m^{4}n^{5}\sqrt{5mn}\)
We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify \(\sqrt{25}+\sqrt{144}\) we must simplify each square root separately first, then add to get the sum of 17.
The expression \(\sqrt{17}+\sqrt{7}\) cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.
In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.
Example \(\PageIndex{19}\)
Simplify: \(3+\sqrt{32}\).
- Answer
-
\[\begin{array}{ll} {}&{3+\sqrt{32}}\\ {\text{Rewrite the radicand as a product using the largest perfect square factor.}}&{3+\sqrt{16·2}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{3+\sqrt{16}·\sqrt{2}}\\ {\text{Simplify.}}&{3+4\sqrt{2}}\\ \end{array}\]
The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!
Example \(\PageIndex{20}\)
Simplify: \(5+\sqrt{75}\).
- Answer
-
\(5+5\sqrt{3}\)
Example \(\PageIndex{21}\)
Simplify: \(2+\sqrt{98}\).
- Answer
-
\(2+7\sqrt{2}\)
The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.
Example \(\PageIndex{22}\)
Simplify: \(\frac{4−\sqrt{48}}{2}\).
- Answer
-
\[\begin{array}{ll} {}&{\frac{4−\sqrt{48}}{2}}\\ {\text{Rewrite the radicand as a product using thelargest perfect square factor.}}&{\frac{4−\sqrt{16·3}}{2}}\\ {\text{Rewrite the radical as the product of two radicals.}}&{\frac{4−\sqrt{16}·\sqrt{3}}{2}}\\ {\text{Simplify.}}&{\frac{4−4\sqrt{3}}{2}}\\ {\text{Factor the common factor from thenumerator.}}&{\frac{4(1−\sqrt{3})}{2}}\\ {\text{Remove the common factor, 2, from thenumerator and denominator.}}&{2(1−\sqrt{3})}\\ \end{array}\]
Example \(\PageIndex{23}\)
Simplify: \(\frac{10−\sqrt{75}}{5}\).
- Answer
-
\(2−\sqrt{3}\)
Example \(\PageIndex{24}\)
Simplify: \(\frac{6−\sqrt{45}}{3}\).
- Answer
-
\(2−\sqrt{5}\)
Use the Quotient Property to Simplify Square Roots
Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.
Example \(\PageIndex{25}\)
Simplify: \(\sqrt{\frac{9}{64}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{9}{64}}}\\ {\text{Since} (\frac{3}{8})^2}&{\frac{3}{8}}\\ \end{array}\]
Example \(\PageIndex{26}\)
Simplify: \(\sqrt{\frac{25}{16}}\).
- Answer
-
\(\frac{5}{4}\)
Example \(\PageIndex{27}\)
Simplify: \(\sqrt{\frac{49}{81}}\).
- Answer
-
\(\frac{7}{9}\)
If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
Example \(\PageIndex{28}\)
Simplify: \(\sqrt{\frac{45}{80}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{45}{80}}}\\ {\text{Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.}}&{\sqrt{\frac{5·9}{5·16}}}\\ {\text{Simplify the fraction by removing common factors.}}&{\sqrt{\frac{9}{16}}}\\ {\text{Simplify.} (\frac{3}{4})^2 =\frac{9}{16}}&{\frac{3}{4}}\\ \end{array}\]
Example \(\PageIndex{29}\)
Simplify: \(\sqrt{\frac{75}{48}}\).
- Answer
-
\(\frac{5}{4}\)
Example \(\PageIndex{30}\)
Simplify: \(\sqrt{\frac{98}{162}}\).
- Answer
-
\(\frac{7}{9}\)
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, \(\frac{a^m}{a^n} = a^{m-n}\), \(a \ne 0\).
Example \(\PageIndex{31}\)
Simplify: \(\sqrt{\frac{m^6}{m^4}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{m^6}{m^4}}}\\ {\text{Simplify the fraction inside the radical first}}&{}\\ {}&{\sqrt{m^2}}\\ {\text{Divide the like bases by subtracting the exponents.}}&{}\\ {\text{Simplify.}}&{m}\\ \end{array}\]
Example \(\PageIndex{32}\)
Simplify: \(\sqrt{\frac{a^8}{a^6}}\).
- Answer
-
a
Example \(\PageIndex{33}\)
Simplify: \(\sqrt{\frac{x^{14}}{x^{10}}}\).
- Answer
-
\(x^2\)
Example \(\PageIndex{34}\)
Simplify: \(\sqrt{\frac{48p^7}{3p^3}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{48p^7}{3p^3}}}\\ {\text{Simplify the fraction inside the radical first.}}&{\sqrt{16p^4}}\\ {\text{Simplify.}}&{4p^2}\\ \end{array}\]
Example \(\PageIndex{35}\)
Simplify: \(\sqrt{\frac{75x^5}{3x}}\).
- Answer
-
\(5x^2\)
Example \(\PageIndex{36}\)
Simplify: \(\sqrt{\frac{72z^{12}}{2z^{10}}}\).
- Answer
-
6z
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
\((\frac{a}{b})^m=\frac{a^{m}}{b^{m}}\), \( b \ne 0\)
We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.
Definition: QUOTIENT PROPERTY OF SQUARE ROOTS
If a, b are non-negative real numbers and \(b \ne 0\), then
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)
Example \(\PageIndex{37}\)
Simplify: \(\sqrt{\frac{21}{64}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{21}{64}}}\\ {\text{We cannot simplify the fraction inside the radical. Rewrite using the quotient property.}}&{\frac{\sqrt{21}}{\sqrt{64}}}\\ {\text{Simplify the square root of 64. The numerator cannot be simplified.}}&{\frac{\sqrt{21}}{8}}\\ \end{array}\]
Example \(\PageIndex{38}\)
Simplify: \(\sqrt{\frac{19}{49}}\).
- Answer
-
\(\frac{\sqrt{19}}{7}\)
Example \(\PageIndex{39}\)
Simplify:\(\sqrt{\frac{28}{81}}\)
- Answer
-
\(\frac{2\sqrt{7}}{9}\)
How to Use the Quotient Property to Simplify a Square Root
Example \(\PageIndex{40}\)
Simplify: \(\sqrt{\frac{27m^3}{196}}\).
- Answer
-
Example \(\PageIndex{41}\)
Simplify: \(\sqrt{\frac{24p^3}{49}}\)
- Answer
-
\(\frac{2p\sqrt{6p}}{7}\)
Example \(\PageIndex{42}\)
Simplify: \(\sqrt{\frac{48x^5}{100}}\)
- Answer
-
\(\frac{2x^2\sqrt{3x}}{5}\)
Definition: SIMPLIFY A SQUARE ROOT USING THE QUOTIENT PROPERTY.
- Simplify the fraction in the radicand, if possible.
- Use the Quotient Property to rewrite the radical as the quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.
Example \(\PageIndex{43}\)
Simplify: \(\sqrt{\frac{45x^5}{y^4}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{45x^5}{y^4}}}\\ {\text{We cannot simplify the fraction inside the radical. Rewrite using the quotient property.}}&{\frac{\sqrt{45x^5}}{\sqrt{y^4}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{9x^4}\sqrt{5x}}{y^2}}\\ {\text{Simplify.}}&{\frac{3x^2\sqrt{5x}}{y^2}}\\ \end{array}\]
Example \(\PageIndex{44}\)
Simplify: \(\sqrt{\frac{80m^3}{n^6}}\)
- Answer
-
\(\frac{4m\sqrt{5m}}{n^3}\)
Example \(\PageIndex{45}\)
Simplify: \(\sqrt{\frac{54u^7}{v^8}}\).
- Answer
-
\(\frac{3u^3\sqrt{6u}}{v^4}\)
Be sure to simplify the fraction in the radicand first, if possible.
Example \(\PageIndex{46}\)
Simplify: \(\sqrt{\frac{81d^9}{25d^4}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{81d^9}{25d^4}}}\\ {\text{Simplify the fraction in the radicand.}}&{\sqrt{\frac{81d^5}{25}}}\\ {\text{Rewrite using the quotient property.}}&{\frac{\sqrt{81d^5}}{\sqrt{25}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{81d^4}\sqrt{d}}{5}}\\ {\text{Simplify.}}&{\frac{9d^2\sqrt{d}}{5}}\\ \end{array}\]
Example \(\PageIndex{47}\)
Simplify: \(\sqrt{\frac{64x^7}{9x^3}}\).
- Answer
-
\(\frac{8x^2}{3}\)
Example \(\PageIndex{48}\)
Simplify: \(\sqrt{\frac{16a^9}{100a^5}}\).
- Answer
-
\(\frac{2a^2}{5}\)
Example \(\PageIndex{49}\)
Simplify: \(\sqrt{\frac{18p^5q^7}{32pq^2}}\).
- Answer
-
\[\begin{array}{ll} {}&{\sqrt{\frac{18p^5q^7}{32pq^2}}}\\ {\text{Simplify the fraction in the radicand.}}&{\sqrt{\frac{9p^4q^5}{16}}}\\ {\text{Rewrite using the quotient property.}}&{\frac{\sqrt{9p^4q^5}}{\sqrt{16}}}\\ {\text{Simplify the radicals in the numerator and the denominator.}}&{\frac{\sqrt{9p^4q^4}\sqrt{q}}{4}}\\ {\text{Simplify.}}&{\frac{3p^2q^2\sqrt{q}}{4}}\\ \end{array}\]
ExAMPLe \(\PageIndex{50}\)
Simplify: \(\sqrt{\frac{50x^5y^3}{72x^4y}}\).
- Answer
-
\(\frac{5y\sqrt{x}}{6}\)
Example \(\PageIndex{51}\)
Simplify: \(\sqrt{\frac{48m^7n^2}{125m^5n^9}}\).
- Answer
-
\(\frac{4m\sqrt{3}}{5n^3\sqrt{5n}}\)
Key Concepts
- Simplified Square Root \(\sqrt{a}\) is considered simplified if a has no perfect-square factors.
- Product Property of Square Roots If a, b are non-negative real numbers, then
\(\sqrt{ab}=\sqrt{a}·\sqrt{b}\)
- Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
- Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.
- Use the product rule to rewrite the radical as the product of two radicals.
- Simplify the square root of the perfect square.
- Quotient Property of Square Roots If a, b are non-negative real numbers and \(b \ne 0\), then
\(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)
- Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
- Simplify the fraction in the radicand, if possible.
- Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.
