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9.3: Simplifying Square Root Expressions


To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect Squares

Rea numbers that are squares of rational numbers are called perfect squares. The numbers (25) and (dfrac{1}{4}) are examples of perfect squares since (25 = 5^2) and (dfrac{1}{4} = (dfrac{1}{2})^2), and (5) and (dfrac{1}{2}) are rational numbers. The number (2) is not a perfect square since (2 = (sqrt{2})^2) and (sqrt{2}) is not a rational number.

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Note

Any indicated square root whose radicand is not a perfect square number is an irrational number.

The numbers (sqrt{6}, sqrt{15}) and (sqrt{dfrac{3}{4}}) are each irrational since each radicand (6, 15, dfrac{3}{4}) is not a perfect square.

The Product Property of Square Roots

Notice that

(egin{array}{flushleft}
sqrt{9 cdot 4} &= sqrt{36} &= 6 & ext{ and }
sqrt{9} sqrt{4} &= 3 cdot 2 &= 6
end{array})

The Product Property (sqrt{xy} = sqrt{x} sqrt{y})

This suggests that in general, if (x) and (y) are positive real numbers,

(sqrt{xy} = sqrt{x} sqrt{y})

The square root of the product is the product of the square roots.

The Quotient Property of Square Roots

We can suggest a similar rule for quotients. Notice that

(sqrt{dfrac{36}{4}} = sqrt{9} = 3) and

(dfrac{sqrt{36}}{sqrt{4}} = dfrac{6}{2} = 3).

Since both (dfrac{36}{4}) and (dfrac{sqrt{36}}{sqrt{4}}) equal (3), it must be that

(sqrt{dfrac{36}{4}} = dfrac{sqrt{36}}{sqrt{4}})

The Quotient Property (sqrt{dfrac{x}{y}} = dfrac{sqrt{x}}{sqrt{y}})

This suggests that in general, if (x) and (y) are positive real numbers,

(sqrt{dfrac{x}{y}} = dfrac{sqrt{x}}{sqrt{y}}, y ot = 0).

The square root of the quotient is the quotient of the square roots.

CAUTION

It is extremely important to remeber that

(sqrt{x + y} ot = sqrt{x} + sqrt{y}) or (sqrt{x - y} ot = sqrt{x} - sqrt{y})

For example, notice that (sqrt{16 + 9} = sqrt{25} = 5), but (sqrt{16} + sqrt{9} = 4 + 3 = 7)

We shall study the process of simplifying a square root expresion by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square Roots Not Involving Fractions

A square root that does not involve fractions is in the simplified form if there is no perfect square in the radicand.

The square roots (sqrt{x},. sqrt{ab}, sqrt{5mn}, sqrt{2(a+5)}) are in simplified form since none of the radicands contains a perfect square.

The square roots (sqrt{x^2}, sqrt{a^3}=sqrt{a^2a}) are not in simplified form since each radicand contains a perfect square.

To simplify a square root expression that does not involve a fraction, we can use the following two rules:

Simplifying Square Roots Without Fractions

  1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
  2. If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

Sample Set A

Simplify each square root.

Example (PageIndex{1})

(sqrt{a^4}). The exponent is even: (dfrac{4}{2} = 2). The exponent on the square root is (2).

(sqrt{a^4} = a^2)

Example (PageIndex{2})

(sqrt{a^6b^{10}}). Both exponents are even: (dfrac{6}{2} = 3) and (dfrac{10}{2} = 5). The exponent on the square root of (a^6) is (3). The exponent on the square root if (b^{10}) is (5).

(sqrt{a^6gb^{10}} = a^3b^5)

Example (PageIndex{3})

(sqrt{y^5}). The exponent is odd: (y^5 = y^4y). The

(sqrt{y^5} = sqrt{y^4y} = sqrt{y^4} sqrt{y} = y^2 sqrt{y})

Example (PageIndex{4})

(egin{array}{flushleft}
sqrt{36a^7b^{11}c^{20}} &= sqrt{6^2a^6ab^{10}bc^{20}} & a^7 = a^6a, b^{11} = b^{10}b
&= sqrt{6^2a^6b^{10}c^{20} cdot ab} & ext{ by the commutative property of multiplication }
&= sqrt{6^2a^6b^{10}c^{20}} sqrt{ab} & ext{ by the product property of square roots }
&= 6a^3b^5c^{10} sqrt{ab}
end{array})

Example (PageIndex{5})

(egin{array}{flushleft}
sqrt{49x^8y^3(a-1)^6} &= sqrt{7^2x^8y^2y(a-1)^6}
&= sqrt{7^2x^8y^2(a-1)^6} sqrt{y}
&= 7x^4y(a-1)^3 sqrt{y}
end{array})

Example (PageIndex{6})

(sqrt{75} = sqrt{25 cdot 3} = sqrt{5^2 cdot 3}= sqrt{5^2} sqrt{3} = 5 sqrt{3})

Practice Set A

Simplify each square root.

Practice Problem (PageIndex{1})

(sqrt{m^8})

Answer

(m^4)

Practice Problem (PageIndex{2})

(sqrt{h^{14}k^{22}})

Answer

(h^7k^{11})

Practice Problem (PageIndex{3})

(sqrt{81a^{12}b^6c^{38}})

Answer

(9a^6b^3c^{19})

Practice Problem (PageIndex{4})

(sqrt{144x^4y^{80}(b+5)^{16}})

Answer

(12x^2y^{40}(b+5)^8)

Practice Problem (PageIndex{5})

(sqrt{w^5})

Answer

(w^2 sqrt{w})

Practice Problem (PageIndex{6})

(sqrt{w^7z^3k^{13}})

Answer

(w^3zk^6 sqrt{wzk})

Practice Problem (PageIndex{7})

(sqrt{27a^3b^4c^5d^6})

Answer

(3ab^2c^2d^3 sqrt{3ac})

Practice Problem (PageIndex{8})

(sqrt{180m^4n^{15}9a-12)^{15}})

Answer

(6m^2n^7(a-12)^7 sqrt{5n(a-12)})

Square Roots Involving Fractions

A square root expression is in simplified form if there are

  1. no perfect squares in the radicand,
  2. no fractions in the radicand, or
  3. no square root expressions in the denominator.

The square root expressions (sqrt{5a}, dfrac{4sqrt{3xy}}{5}), and (dfrac{11m^2n sqrt{a-4}}{2x^2}) are in simplified form

The square root expressions (sqrt{dfrac{3x}{8}}, sqrt{dfrac{4a^4b^3}{5}}), and (dfrac{2y}{sqrt{3x}}) are not in simplified form.

Simplifying Square Roots with Fractions

To simplify the square root expression (sqrt{dfrac{x}{y}}),

  1. Write the expression as (dfrac{sqrt{x}}{sqrt{y}}) using the rule (sqrt{dfrac{x}{y}} = dfrac{sqrt{x}}{sqrt{y}}).
  2. Multiply the fraction by 1 in the form of (dfrac{sqrt{y}}{sqrt{y}}).
  3. Simplify the remaining fraction, (dfrac{sqrt{xy}}{y}).

Rationalizing the Denominator

The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that ((sqrt{y})(sqrt{y}) = y).

Sample Set B

Simplify each square root.

Example (PageIndex{7})

(sqrt{dfrac{9}{25}} = dfrac{sqrt{9}}{sqrt{25}} = dfrac{3}{5})

Example (PageIndex{8})

(sqrt{dfrac{3}{5}}=dfrac{sqrt{3}}{sqrt{5}}=dfrac{sqrt{3}}{sqrt{5}} cdot dfrac{sqrt{5}}{sqrt{5}}=dfrac{sqrt{15}}{5})

Example (PageIndex{9})

(sqrt{dfrac{9}{8}}=dfrac{sqrt{9}}{sqrt{8}}=dfrac{sqrt{9}}{sqrt{8}} cdot dfrac{sqrt{8}}{sqrt{8}}=dfrac{3 sqrt{8}}{8}=dfrac{3 sqrt{4 cdot 2}}{8}=dfrac{3 sqrt{4} sqrt{2}}{8}=dfrac{3 cdot 2 sqrt{2}}{8}=dfrac{3 sqrt{2}}{4})

Example (PageIndex{10})

(sqrt{dfrac{k^{2}}{m^{3}}}=dfrac{sqrt{k^{2}}}{sqrt{m^{3}}}=dfrac{k}{sqrt{m^{3}}}=dfrac{k}{sqrt{m^{2} m}}=dfrac{k}{sqrt{m^{2} sqrt{m}}}=dfrac{k}{m sqrt{m}}=dfrac{k}{m sqrt{m}} cdot dfrac{sqrt{m}}{sqrt{m}}=dfrac{k sqrt{m}}{m sqrt{m} sqrt{m}}=dfrac{k sqrt{m}}{m cdot m}=dfrac{k sqrt{m}}{m^{2}})

Example (PageIndex{11})

(egin{array}{flushleft}
sqrt{x^2 - 8x + 16} &= sqrt{(x-4)^2}
&= x-4
end{array})

Practice Set B

Simplify each square root.

Practice Problem (PageIndex{9})

(sqrt{dfrac{81}{25}})

Answer

(dfrac{9}{5})

Practice Problem (PageIndex{10})

(sqrt{dfrac{2}{7}})

Answer

(dfrac{sqrt{14}}{7})

Practice Problem (PageIndex{11})

(sqrt{dfrac{4}{5}})

Answer

(dfrac{2 sqrt{5}}{5})

Practice Problem (PageIndex{12})

(sqrt{dfrac{10}{4}})

Answer

(dfrac{sqrt{10}}{2})

Practice Problem (PageIndex{13})

(sqrt{dfrac{9}{4}})

Answer

(dfrac{3}{2})

Practice Problem (PageIndex{14})

(sqrt{dfrac{a^3}{6}})

Answer

(dfrac{a sqrt{6a}}{6})

Practice Problem (PageIndex{15})

(sqrt{dfrac{y^4}{x^3}})

Answer

(dfrac{y^2 sqrt{x}}{x^2})

Practice Problem (PageIndex{16})

(sqrt{dfrac{32a^5}{b^7}})

Answer

(dfrac{4a^2 sqrt{2ab}}{b^4})

Practice Problem (PageIndex{17})

(sqrt{(x+9)^2})

Answer

(x+9)

Practice Problem (PageIndex{18})

(sqrt{x^2 + 14x + 49})

Answer

(x+7)

Exercises

For the following problems, simplify each of the radical expressions.

Exercise (PageIndex{1})

(sqrt{8b^2})

Answer

(2b sqrt{2})

Exercise (PageIndex{2})

(sqrt{20a^2})

Exercise (PageIndex{3})

(sqrt{24x^4})

Answer

(2x^2 sqrt{6})

Exercise (PageIndex{4})

(sqrt{27y^6})

Exercise (PageIndex{5})

(sqrt{a^5})

Answer

(a^2sqrt{a})

Exercise (PageIndex{6})

(sqrt{m^7})

Exercise (PageIndex{7})

(sqrt{x^{11}})

Answer

(x^5 sqrt{x})

Exercise (PageIndex{8})

(sqrt{y^{17}})

Exercise (PageIndex{9})

(sqrt{36n^9})

Answer

(6n^4 sqrt{n})

Exercise (PageIndex{10})

(sqrt{49x^{13}})

Exercise (PageIndex{11})

(sqrt{100x^5y^{11}})

Answer

(10x^2y^5 sqrt{xy})

Exercise (PageIndex{12})

(sqrt{64a^7b^3})

Exercise (PageIndex{13})

(5 sqrt{16m^6n^7})

Answer

(20m^3n^3 sqrt{n})

Exercise (PageIndex{14})

(8 sqrt{9a^4b^{11}})

Exercise (PageIndex{15})

(3 sqrt{16x^3})

Answer

(12x sqrt{x})

Exercise (PageIndex{16})

(8 sqrt{25y^3})

Exercise (PageIndex{17})

(sqrt{12a^4})

Answer

(2a^2 sqrt{3})

Exercise (PageIndex{18})

(sqrt{32x^7})

Answer

(4x^3 sqrt{2x})

Exercise (PageIndex{19})

(sqrt{12y^{13}})

Exercise (PageIndex{20})

(sqrt{50a^3b^9})

Answer

(5ab^4 sqrt{2ab})

Exercise (PageIndex{21})

(sqrt{48p^{11}q^5})

Exercise (PageIndex{22})

(4 sqrt{18a^5b^{17}})

Answer

(12a^2b^8 sqrt{2ab})

Exercise (PageIndex{23})

(8 sqrt{108x^{21}y^3})

Exercise (PageIndex{24})

(-4 sqrt{75a^4b^6})

Answer

(-20a^2b^3 sqrt{3})

Exercise (PageIndex{25})

(-6 sqrt{72x^2y^4z^{10}})

Exercise (PageIndex{26})

(-sqrt{b^{12}})

Answer

(-b^6)

Exercise (PageIndex{27})

(- sqrt{c^{18}})

Exercise (PageIndex{28})

(sqrt{a^2b^2c^2})

Answer

(abc)

Exercise (PageIndex{29})

(sqrt{4x^2y^2z^2})

Exercise (PageIndex{30})

(- sqrt{9a^2b^3})

Answer

(-3ab sqrt{b})

Exercise (PageIndex{31})

(- sqrt{16x^4y^5})

Exercise (PageIndex{32})

(sqrt{m^6n^8p^{12}q^{20}})

Answer

(m^3n^4p^6q^{10})

Exercise (PageIndex{33})

(sqrt{r^2})

Exercise (PageIndex{34})

(sqrt{p^2})

Answer

(p)

Exercise (PageIndex{35})

(sqrt{dfrac{1}{4}})

Exercise (PageIndex{36})

(sqrt{dfrac{1}{16}})

Answer

(dfrac{1}{4})

Exercise (PageIndex{37})

(sqrt{dfrac{4}{25}})

Exercise (PageIndex{38})

(sqrt{dfrac{9}{49}})

Answer

(dfrac{3}{7})

Exercise (PageIndex{39})

(dfrac{5 sqrt{8}}{sqrt{3}})

Exercise (PageIndex{40})

(dfrac{2 sqrt{32}}{sqrt{3}})

Answer

(dfrac{8 sqrt{6}}{3})

Exercise (PageIndex{41})

(sqrt{dfrac{5}{6}})

Exercise (PageIndex{42})

(sqrt{dfrac{2}{7}})

Answer

(dfrac{sqrt{14}}{7})

Exercise (PageIndex{43})

(sqrt{dfrac{3}{10}})

Exercise (PageIndex{44})

(sqrt{dfrac{4}{3}})

Answer

(dfrac{2 sqrt{3}}{3})

Exercise (PageIndex{45})

(-sqrt{dfrac{2}{5}})

Exercise (PageIndex{46})

(-sqrt{dfrac{3}{10}})

Answer

(-dfrac{sqrt{30}}{10})

Exercise (PageIndex{47})

(sqrt{dfrac{16a^2}{5}})

Exercise (PageIndex{48})

(sqrt{dfrac{24a^5}{7}})

Answer

(dfrac{2a^2 sqrt{42a}}{7})

Exercise (PageIndex{49})

(sqrt{dfrac{72x^2y^3}{5}})

Exercise (PageIndex{50})

(sqrt{dfrac{2}{a}})

Answer

(dfrac{sqrt{2a}}{a})

Exercise (PageIndex{51})

(sqrt{dfrac{5}{b}})

Exercise (PageIndex{52})

(sqrt{dfrac{6}{x^3}})

Answer

(dfrac{sqrt{6x}}{x^2})

Exercise (PageIndex{53})

(sqrt{dfrac{12}{y^5}})

Exercise (PageIndex{54})

(sqrt{dfrac{49x^2y^5z^9}{25a^3b^{11}}})

Answer

(dfrac{7 x y^{2} z^{4} sqrt{a b y z}}{5 a^{2} b^{6}})

Exercise (PageIndex{55})

(sqrt{dfrac{27 x^{6} y^{15}}{3^{3} x^{3} y^{5}}})

Exercise (PageIndex{56})

(sqrt{(b+2)^4})

Answer

((b+2)^2)

Exercise (PageIndex{57})

(sqrt{(a-7)^8})

Exercise (PageIndex{58})

(sqrt{(x+2)^6})

Answer

((x+2)^3)

Exercise (PageIndex{59})

(sqrt{(x+2)^2(x+1)^2})

Exercise (PageIndex{60})

(sqrt{(a-3)^4(a-1)^2})

Answer

((a-3)^2(a-1))

Exercise (PageIndex{61})

(sqrt{(b+7)^8(b-7)^6})

Exercise (PageIndex{62})

(sqrt{a^2 - 10a + 25})

Answer

((a-5))

Exercise (PageIndex{63})

(sqrt{b^2 + 6b + 9})

Exercise (PageIndex{64})

(sqrt{(a^2 - 2a + 1)^4})

Answer

((a-1)^4)

Exercise (PageIndex{65})

(sqrt{(x^2 + 2x + 1)^{12}})

Exercises For Review

Exercise (PageIndex{66})

Solve the inequality (3(a + 2) le 2(3a + 4))

Answer

(a ge -dfrac{2}{3})

Exercise (PageIndex{67})

Graph the inequality (6x le 5(x+1) - 6)

Exercise (PageIndex{68})

Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

Answer

positive; negative

Exercise (PageIndex{69})

Simplify the complex fraction: (dfrac{5+frac{1}{x}}{5-frac{1}{x}})

Exercise (PageIndex{70})

Simplify (sqrt{121x^4w^6z^8}) by removing the radical sign.

Answer

(11x^2w^3z^4)


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Simplifying Expressions With Square Roots

Remember that when a number (n) is multiplied by itself, we write (^<2>) and read it “n squared.” For example, (<15>^<2>) reads as “15 squared,” and 225 is called the square of 15, since (<15>^<2>=225) .

Square of a Number

If (^<2>=m) , then (m) is the square of (n) .

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is (m) is called a square root of (m) .

Square Root of a Number

If (^<2>=m) , then (n) is a square root of (m) .

Notice (^<2>=225) also, so (-15) is also a square root of 225. Therefore, both 15 and (-15) are square roots of 225.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, (sqrt) , denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because (<0>^<2>=0) , (sqrt<0>=0) . Notice that zero has only one square root.

Square Root Notation

(sqrt) is read as “the square root of (m) .”

The square root of (m) , (sqrt) , is the positive number whose square is (m) .

Since 15 is the positive square root of 225, we write (sqrt<225>=15) . Fill in the figure below to make a table of square roots you can refer to as you work this tutorial.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write (sqrt<225>=15) . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, ( ext<−>sqrt<225>=-15) .


By Removing Perfect Square Factors

“Simplifying a square root” means rewriting it as an expression of the same value, but with the number or expression inside the square root as small or simple as possible.

We will illustrate the technique here for square roots involving just numbers, but this method is most important in simplifying square roots containing algebraic expressions.

As an example, notice that we can do the following:

Thus has the same value as . But, we would consider to be a simpler form because the quantity in the square root is a smaller number. If we rewrite the above example with the steps in reverse order, we can see the strategy for simplifying a square root when that is possible.

If possible, separate or factor 45 into a product of two numbers, one of which is the square of a whole number. (Recall, we called such numbers “perfect squares” earlier.)
Use the rule for multiplying two square roots.
since the square root of a square is the original number.
The multiplication symbol can be omitted.

Since the remaining number in the square root, the 5, obviously cannot be written as a product of a perfect square and another number, we have achieved as much simplification here as is possible.

This strategy for simplifying square root expressions requires us to develop a strategy for deducing how numbers can be rewritten as a product involving one or more perfect squares – indeed, we need to be able to rewrite the original number in the square root as a product of perfect squares, and the one smallest value which is not a perfect square.


Example: How To Use the Product Property to Simplify a Square Root

Solution

Notice in the previous example that the simplified form of (sqrt<50>) is (5sqrt<2>) , which is the product of an integer and a square root. We always write the integer in front of the square root.


9.3: Simplifying Square Root Expressions

Before you get started, take this readiness quiz.

  1. Simplify: ⓐ 9 2 9 2 ⓑ ( 𕒽 ) 2 ( 𕒽 ) 2 ⓒ − 9 2 − 9 2 .
    If you missed this problem, review [link].
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review [link].
  3. For each number, identify whether it is a real number or not a real number:
    ⓐ − 100 − 100 ⓑ � � .
    If you missed this problem, review [link].

Simplify Expressions with Square Roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is m m is called a square root of m m .

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, m m , denotes the positive square root. The positive square root is also called the principal square root.

We also use the radical sign for the square root of zero. Because 0 2 = 0 0 2 = 0 , 0 = 0 0 = 0 . Notice that zero has only one square root.

Since 15 is the positive square root of 225, we write 225 = 15 225 = 15 . Fill in [link] to make a table of square roots you can refer to as you work this chapter.

When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.

Notice the different answers in parts ⓐ and ⓑ !

Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at [link] below.

Number Square Root
4 4 4 = 2
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9 = 3

The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that 5 5 must be between 2 and 3. Using inequality symbols, we write:

Think of the perfect square numbers closest to 60. Make a small table of these perfect squares and their squares roots.

Locate 60 between two consecutive perfect squares.
60 60 is between their square roots.

Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the x x key on your calculator. You will use this key to approximate square roots.

When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is ≈ ≈ and it is read ‘approximately.’

Suppose your calculator has a 10-digit display. You would see that

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to 5, but are not exactly equal to 5.

Using the square root key on a calculator and then rounding to two decimal places, we can find:


Simplify the expression and express the answer using rational exponents. Assume that x and y denote… Show more Simplify the expression and express the answer using rational exponents. Assume that x and y denote positive numbers. ^3(square root symbol) 512x^2y^4 / 8x^5y *Note: the power of 3 is in front of the square root symbol. The entire fraction is inside of the square root symbol. • Show less

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Solution

Solution

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.

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.

Quotient Property of Square Roots

If a, b are non-negative real numbers and (b e 0) , then


Watch the video: Rewriting Radical Expressions (October 2021).