9.2: Simplify Square Roots
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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 √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 √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.
√a is considered simplified if a has no perfect square factors.
So √31 is simplified. But √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=ambm. The corresponding property of square roots says that √ab=√a·√b.
If a, b are non-negative real numbers, then √ab=√a·√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
Simplify: √50.
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Simplify: √48.
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4√3
Simplify: √45.
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3√5
Notice in the previous example that the simplified form of √50 is 5√2, which is the product of an integer and a square root. We always write the integer in front of the square root.
- 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.
Simplify: √500.
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√500Rewrite the radicand as a product using the largest perfect square factor√100·5Rewrite the radical as the product of two radicals√100·√5Simplify10√5
Simplify: √288.
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12√2
Simplify:√432.
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12√3
We could use the simplified form 10√5 to estimate √500. We know √5 is between 2 and 3, and √500 is 10√5. So √500 is between 20 and 30.
The next example is much like the previous examples, but with variables.
Simplify: √x3.
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√x3Rewrite the radicand as a product using the largest perfect square factor√x2·xRewrite the radical as the product of two radicals√x2·√xSimplifyx√x
Simplify:√b5.
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b2√b
Simplify: √p9.
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p4√p
We follow the same procedure when there is a coefficient in the radical, too.
Simplify: √25y5.
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√25y5Rewrite the radicand as a product using the largest perfect square factor.√25y4·yRewrite the radical as the product of two radicals.√25y4·√ySimplify.5y2√y
Simplify: √16x7.
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4x3√x
Simplify: √49v9.
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7v4√v
In the next example both the constant and the variable have perfect square factors.
Simplify: √72n7.
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√72n7Rewrite the radicand as a product using the largest perfect square factor.√36n6·2nRewrite the radical as the product of two radicals.√36n6·√2nSimplify.6n3√2n
Simplify: √32y5.
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4y2√2y
Simplify: √75a9.
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5a4√3a
Simplify: √63u3v5.
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√63u3v5Rewrite the radicand as a product using the largest perfect square factor.√9u2v4·7uvRewrite the radical as the product of two radicals.√9u2v4·√7uvSimplify.3uv2√7uv
Simplify: √98a7b5.
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7a3b2√2ab
Simplify: √180m9n11.
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6m4n5√5mn
We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify √25+√144 we must simplify each square root separately first, then add to get the sum of 17.
The expression √17+√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.
Simplify: 3+√32.
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3+√32Rewrite the radicand as a product using the largest perfect square factor.3+√16·2Rewrite the radical as the product of two radicals.3+√16·√2Simplify.3+4√2
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!
Simplify: 5+√75.
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5+5√3
Simplify: 2+√98.
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2+7√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.
Simplify: 4−√482.
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4−√482Rewrite the radicand as a product using thelargest perfect square factor.4−√16·32Rewrite the radical as the product of two radicals.4−√16·√32Simplify.4−4√32Factor the common factor from thenumerator.4(1−√3)2Remove the common factor, 2, from thenumerator and denominator.2(1−√3)
Simplify: 10−√755.
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2−√3
Simplify: 6−√453.
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2−√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.
Simplify: √964.
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√964Since(38)238
Simplify: √2516.
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54
Simplify: √4981.
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79
If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
Simplify: √4580.
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√4580Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.√5·95·16Simplify the fraction by removing common factors.√916Simplify.(34)2=91634
Simplify: √7548.
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54
Simplify: √98162.
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79
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, aman=am−n, a≠0.
Simplify: √m6m4.
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√m6m4Simplify the fraction inside the radical first√m2Divide the like bases by subtracting the exponents.Simplify.m
Simplify: √a8a6.
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a
Simplify: √x14x10.
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x2
Simplify: √48p73p3.
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√48p73p3Simplify the fraction inside the radical first.√16p4Simplify.4p2
Simplify: √75x53x.
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5x2
Simplify: √72z122z10.
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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.
(ab)m=ambm, b≠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.
If a, b are non-negative real numbers and b≠0, then
√ab=√a√b
Simplify: √2164.
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√2164We cannot simplify the fraction inside the radical. Rewrite using the quotient property.√21√64Simplify the square root of 64. The numerator cannot be simplified.√218
Simplify: √1949.
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√197
Simplify:√2881
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2√79
How to Use the Quotient Property to Simplify a Square Root
Simplify: √27m3196.
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Simplify: √24p349
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2p√6p7
Simplify: √48x5100
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2x2√3x5
- 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.
Simplify: √45x5y4.
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√45x5y4We cannot simplify the fraction inside the radical. Rewrite using the quotient property.√45x5√y4Simplify the radicals in the numerator and the denominator.√9x4√5xy2Simplify.3x2√5xy2
Simplify: √80m3n6
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4m√5mn3
Simplify: √54u7v8.
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3u3√6uv4
Be sure to simplify the fraction in the radicand first, if possible.
Simplify: √81d925d4.
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√81d925d4Simplify the fraction in the radicand.√81d525Rewrite using the quotient property.√81d5√25Simplify the radicals in the numerator and the denominator.√81d4√d5Simplify.9d2√d5
Simplify: √64x79x3.
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8x23
Simplify: √16a9100a5.
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2a25
Simplify: √18p5q732pq2.
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√18p5q732pq2Simplify the fraction in the radicand.√9p4q516Rewrite using the quotient property.√9p4q5√16Simplify the radicals in the numerator and the denominator.√9p4q4√q4Simplify.3p2q2√q4
Simplify: √50x5y372x4y.
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5y√x6
Simplify: √48m7n2125m5n9.
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4m√35n3√5n
Key Concepts
- Simplified Square Root √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
√ab=√a·√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≠0, then
√ab=√a√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.