4. Operating with Square Roots
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Chapter 8
4.
Operating with Square Roots
This lesson provides a comprehensive guide to understanding and operating with square roots. It covers key concepts such as the product property, which helps in simplifying the multiplication of square roots, and the quotient property, useful for dividing square roots. Additionally, the concept of rationalization is discussed, a technique used to eliminate square roots from the denominator of a fraction. These principles are not just theoretical; they find applications in various scientific and engineering tasks, such as calculating distances, areas, and even in some algorithms in computer science. Overall, the material serves as an invaluable lesson for mastering the complexities of square roots and their practical applications in real-world scenarios.
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| | 18 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Operating with Square Roots
Slide of 18
This lesson will introduce the concept of a square root and how to operate its many properties.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
The Golden Ratio
The golden ratio is the ratio of two positive numbers with the property that their ratio is the same as the ratio of their sum to the larger of the two numbers. It is represented by the Greek letter 
φ.Its value is given by the following expression.
φ=21+5
Not only mathematicians, but throughout history, architects and artists have used this ratio. To this day, people still paint on canvases shaped as a golden rectangle. The length ℓ and width w of this rectangle are in the golden ratio.
wℓ=φ
Dylan, an avid painter, sets out to buy an uncut canvas. Feeling savvy, he thinks he can cut the canvas himself in such a way that it becomes a golden rectangle. Dylan aims to cut the canvas a length ℓ of 60 inches. What should be the width? Discussion
Square Roots
Some numbers cannot be expressed as the ratio of two integers. These numbers have a special name.
Concept
Irrational Numbers
The set of irrational numbers is formed by all numbers that cannot be expressed as the ratio between two integers.
2,3,5,e,π
Irrational numbers are real numbers, but they cannot be expressed as fractions. Also, the decimal expansion of irrational numbers is not repeating and non-terminating.
2π=1.41421356237…=3.14159265359…
In other words, a number is irrational if it is not rational. Although this number set does not have its own symbol, it is sometimes represented with a combination of other symbols. R−QorR ∖ Q
From the examples given above, 2, 3, and 5 are called the square root of 2, the square root of 3, and the square root of 5, respectively.
Concept
Square Root
A square root of a number a is a number that, when multiplied by itself, equals a. For example, 4 and -4 are the square roots of 16.
4⋅4-4⋅(-4)=16=16
All positive numbers have two square roots — one positive and one negative. To avoid ambiguity, when talking about the square root of a number, only the positive root, also known as its principal root, is considered. Furthermore, to denote the square root, the symbol 4is used. For example, the square root of 16 is denoted as
16.
16=4
In the example above, the principal root of 16 is 4 because 4 multiplied by itself equals 16 and 4 is positive. When a number is a perfect square, its square roots are integers. The square roots of positive integers that are non-perfect squares are irrational numbers. | Principal Root of Perfect Squares | Principal Root of Non-Perfect Squares | ||
|---|---|---|---|
| Perfect Square | Principal Root (Integer Number) |
Non-Perfect Square | Principal Root (Irrational Number) |
| 1 | 1=1 | 2 | 2≈1.414213… |
| 4 | 4=2 | 3 | 3≈1.732050… |
| 9 | 9=3 | 5 | 5≈2.236067… |
| 16 | 16=4 | 10 | 10≈3.162277… |
| 25 | 25=5 | 20 | 20≈4.472135… |
Extra
Square Roots of Fractions and Decimal NumbersSeparate from whole numbers, the square roots of fractions can be calculated by taking square roots of the numerator and denominator separately. Consider the following example.
169=4232⇒169=43
The square roots of decimal numbers can be calculated by writing them in the fraction form. Then, the square roots of the numerator and denominator are calculated. Consider the following example.
0.36=10036=10262⇓10036=106=0.6
Example
Fertilizing a Garden Using the Square Root
Emily visited her grandparent's new house for a family gathering. She loves their huge backyard! Her grandpa, eager to let her explore, told her she can use some of the free space and some leftover fertilizer to make herself a little flower garden!
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Hint
What is the square root of 81?
Solution
The area of Emily's garden can be found by using the formula for the area of a square. Further, since there is enough fertilizer to cover 81 square feet, this number can be substituted into the formula.
A=ℓ2 substitute 81=ℓ2
The obtained equation states that ℓ is a number whose square equals 81. This means that ℓ multiplied by itself is 81. Therefore, ℓ is the square root of 81. Since ℓ represents the side length of a square, it must be positive. Because of this fact, only the principal root will be considered.
For the garden's area to be 81 square feet, the side length must be 9 feet. Emily can now start gardening in full confidence! Discussion
Product Property of Square Roots
Sometimes it is necessary to simplify a square root. The Product Property of Square Roots can be helpful when doing so.
Given two non-negative numbers a and b, the square root of their product equals the product of the square root of each number.
ab=a⋅b, for a≥0 and b≥0
Proof
Let x, y, and z be three non-negative numbers such that x=a, y=b, and z=ab. By the definition of a square root, each of these numbers squared is equal to its corresponding radicand.
Since x, y, and z are non-negative, the final equation implies that z=xy.
⎩⎪⎪⎨⎪⎪⎧x2=ay2=bz2=ab(I)(II)(III)
Next, multiply Equation (I) by y2.
x2=⇓x2y2= a ay2
Now, substitute Equations (II) and (III) into this equation.
x2y2=ay2
▼
Substitute values and simplify
Substitute
y2=b
x2y2=ab
Substitute
ab=z2
x2y2=z2
ProdPowII
ambm=(ab)m
(xy)2=z2
RearrangeEqn
Rearrange equation
z2=(xy)2
z2=(xy)2⇒z=xy
The last step is substituting z=ab, x=a, and y=b into this equation.
z=xy⇔ab=a⋅b
Example
Finding the Area of a Plot
At the family gathering, Emily's aunt named Auntie Agent is gushing about her job as a real estate agent. She is bragging about a recent business deal. She purchased a new plot that is located next to two plots she also owns, as highlighted in the diagram.
Auntie Agent wants to resale her newly purchased plot in a few years. To do so, she needs to know the area of the plot. Unfortunately, the land bill is severely faded, and the area is unreadable. Luckily, she knows the areas of the two square plots next to it. Knowing that Emily is good at math, Auntie Agent asks her for help.
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Hint
Use the formula for the area of a square and the formula for the area of a rectangle.
Solution
The area A of the new plot can be found by using the formula for the area of a rectangle.
Finally, the area of the newly acquired plot can be found by substituting these values into the formula for the area of a rectangle. Then, the Product Property of Square Roots can be used.
Therefore, the area of the new plot is 240 square meters. Auntie Agent is ready to wheel and deal!
A=ℓw
Analyzing the diagram, it can be realized that ℓ and w correspond to the lengths of the square plots. Since the areas of the square plots are known, it is possible to find ℓ and w.
| Area of Square Plot | Side Length |
|---|---|
| ℓ2=160 | ℓ=160 |
| w2=360 | w=360 |
A=ℓw
SubstituteII
ℓ=160, w=360
A=160⋅360
ProdSqrt
a⋅b=a⋅b
A=160⋅360
Multiply
Multiply
A=57600
CalcRoot
Calculate root
A=240
Example
Simplifying Square Roots
Auntie Agent finds herself bored of the family gathering. She sneaks off to the kitchen wanting to calculate a few math problems from her kid's math textbook! She notices an interesting expression on a graphing calculator.
She notices that the square root of 8 appears to be twice the value of the square root of 2. Auntie Agent, curious to know why, checks her kid's notes and sees the following notes from his class.
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Hint
Factor 18 using perfect squares.
Solution
In order to use the Product Property of Square Roots, the radicand should be factored using perfect squares.
18=9⋅2
Knowing this, the Product Property of Square Roots can be used.
Therefore, 18 equals 32. Auntie Agent feels relieved to have figured out what the graphing calculator expressed. Pop Quiz
Simplify the Expression
Use the Product Property of Square Roots to simplify the given square roots.
Discussion
Quotient Property of Square Roots
When working with square roots, just like how the product of a square root operates, there is a similar property for quotients.
Let a be a non-negative number and b be a positive number. The square root of the quotient ba equals the quotient of the square roots of a and b.
ba=ba, for a≥0, b>0
Proof
Let x, y, and z be non-negative numbers such that x=a, y=b, and z=ba. By the definition of a square root, each of these numbers squared is equal to its corresponding radicand.
Since x, y, and z are non-negative, the final equation implies that z=yx.
⎩⎪⎪⎪⎨⎪⎪⎪⎧x2=ay2=bz2=ba(I)(II)(III)
Since b is a positive number, y2 is also positive. Therefore, Equation (I) can be divided by y2.
x2=⇓y2x2= a y2a
Now, substitute Equations (II) and (III) into this equation.
y2x2=y2a
▼
Substitute values and simplify
Substitute
y2=b
y2x2=ba
Substitute
ba=z2
y2x2=z2
bmam=(ba)m
(yx)2=z2
RearrangeEqn
Rearrange equation
z2=(yx)2
z2=(yx)2⇒z=yx
The last step is substituting z=ba, x=a, and y=b into this equation.
z=yx⇔ba=ba
Example
Finding the Hypotenuse of a Right Triangle
Emily roams over to see what her cousins are up to, and one of them is working on some geometry homework. They need to find the hypotenuse of the right triangle shown in the diagram.
a2+b2=c2⇓c=a2+b2
After adding the squares of the legs, they are left with a numeric expression for the hypotenuse of the triangle. They wonder if this can be simplified. 
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Hint
Use the Quotient Property of Square Roots.
Solution
Emily and their cousin already did most of the work! In order to simplify the given expression the Quotient Property of Square Roots can be used. Then, the square roots of the numerator and denominator can be calculated.
Therefore, the length of the hypotenuse is 75.
Pop Quiz
Simplify the Square Root
Use the Quotient Property of Square Roots to simplify the given square root.
Discussion
Rationalizing a Denominator
The square root of an irreducible fraction whose denominator is a perfect square will result in a numeric expression with an integer denominator. However, a fraction can have a denominator that is not a perfect square.
63
If the square root of such a fraction is calculated, the denominator will be an irrational number. There is a way of avoiding irrational numbers in a denominator.
When a fraction has a radical denominator that is an irrational number, it is convenient to rewrite the fraction so that the denominator is an integer. This process is known as rationalization. Consider the following example.
63
The numeric expression can be rationalized by multiplying the numerator and denominator by the same factor — a factor that removes the radical from the denominator. The fraction is then simplified, if possible.
1
Multiply by the Denominator
expand_more To remove the radical in the denominator, the numerator and the denominator are both multiplied by the same radical. In this case, multiply by 6.
6⋅63⋅6
2
Simplify the Fraction
expand_more The fraction can now be simplified.
The fraction has now been rationalized because the denominator is an integer number.
Pop Quiz
Rationalizing Denominators
Rationalize the denominator of the given numeric expression.
Discussion
Irrational Conjugates and Rationalization
Some irrational numbers are written as expressions involving rational and irrational terms. As performed previously with perfect squares, these expressions cannot be further simplified into one term and are left written as a sum or difference. Consider the following example.
3+2
Given an irrational number in this form, it is possible to find another irrational number by switching the sign of the irrational term. Concept
Irrational Conjugate
Let a, b, and c be rational numbers, with c irrational. The irrational conjugate of a+bc is obtained by switching the sign of the irrational term.
Numbera+bca−bcswitch signswitch signConjugatea−bca+bc
For example, the conjugate of 2−53 is 2+53.When the denominator of a numeric expression has a number in this form, it can be rationalized by following a standard procedure.
Method
Rationalizing a Denominator Using Conjugates
When a numeric expression has a denominator that is an irrational radical expression, it is convenient to rewrite the expression so that the denominator is a rational number. This process is known as rationalization. Consider the following example.
5+32
Since the denominator involves a rational and a square root, the fraction can be rewritten by multiplying both the numerator and denominator by the irrational conjugate of the denominator. The resulting expression can then be simplified.
1
Multiply by the Conjugate of the Denominator
expand_more To get rid of the radical number in the denominator, the numerator and the denominator are multiplied by its irrational conjugate. In this case, multiply by 5−3.
(5+3)(5−3)2(5−3)
2
Simplify the Expression
expand_more The obtained expression can now be simplified.
The expression has now been rationalized because the denominator is an integer number.
(5+3)(5−3)2(5−3)
ExpandDiffSquares
(a+b)(a−b)=a2−b2
52−(3)22(5−3)
CalcPow
Calculate power
25−(3)22(5−3)
PowSqrt
(a)2=a
25−32(5−3)
SubTerm
Subtract term
222(5−3)
ReduceFrac
ba=b/2a/2
115−3
Example
Simplifying a Ratio that Involves Radicals
Emily now goes over to her cousin Dylan, who looks bored. He says he would rather be painting. She has an idea to cheer him up and shows him the phenomenon of free fall. She walks to the top of the stairs and starts dropping stuff!
9+21+4
What is the simplest way of writing this expression? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{9-3\\sqrt{2}}{7}"]}}
Hint
Rationalize the denominator using the irrational conjugate.
Solution
Note that 1, 4, and 9 are perfect squares. Therefore, their square roots can be calculated.
Since the denominator of this expression is an irrational number, it needs to be rationalized in order to be written in its simplest form. This can be done by multiplying the numerator and denominator by the irrational conjugate of the denominator.
3+23⋅3−23−2
The expression can be now be simplified.
Discussion
Adding and Subtracting Radicals
Some irrational numbers are written as expressions involving the sum of irrational terms. Below is an example.
32+52
Given an expression in this form, it is possible — under certain circumstances — to simplify it into one term. A numeric or algebraic expression that contains two or more radical terms with the same radicand and the same index can be simplified by adding or subtracting the corresponding coefficients.
anx±bnx=(a±b)nx
Here, a,b, and x are real numbers and n is a natural number. If n is even, then x must be greater than or equal to zero.
Proof
If n is an even number, then x is greater than or equal to zero. If n is odd, then x can be any real number. In both cases, nx is a real number. This means that the Distributive Property can be used to factor out nx.
38+22
This expression can be simplified by rewriting 8 in terms of 2. 38+22
▼
Simplify
SplitIntoFactors
Split into factors
34⋅2+22
SqrtProd
a⋅b=a⋅b
342+22
CalcRoot
Calculate root
3⋅22+22
Multiply
Multiply
62+22
62+22
AddTerms
Add terms
82
Example
Using the Properties of Square Roots to Simplify an Expression
Emily's cousins Ali and Davontay want to ace next week's math test. An exercise on their study guide asks them to simplify the following numeric expression.
Help Ali and Davontay simplify the given expression!
24−633+12
Ali says that, since there are no like radicals, the expression cannot be simplified. Davontay suggests the use of the properties of square roots to see if any like radicals appear. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["(5\/2)\\sqrt{2}","2.5\\sqrt{2}","(5\/2)\\sqrt[2]{2}","2.5\\sqrt[2]{2}"]}}
Hint
Use the Product Property of Square Roots to find like radicals. Simplify by rationalizing the denominator.
Solution
In order to add radicals they need to have the same radicand. The Product Property of Square Roots can be used in the numerator and the denominator to find like radicals.
Finally, the denominator will be rationalized.
24−633+12
▼
Simplify numerator
SplitIntoFactors
Split into factors
24−633+4(3)
SqrtProd
a⋅b=a⋅b
24−633+23
AddTerms
Add terms
24−653
▼
Simplify denominator
653
653
▼
Simplify
MovePartNumLeft
ca⋅b=a⋅cb
563
DivSqrt
ba=ba
563
ReduceFrac
ba=b/3a/3
521
SqrtQuot
ba=ba
521
MoveLeftFacToNum
a⋅cb=ca⋅b
25
ExpandFrac
ba=b⋅2a⋅2
2⋅252
MultSqrt
a⋅a=a
252
Closure
Finding the Dimensions of a Golden Rectangle
The challenge presented at the beginning can be solved by using the mathematical tools provided in this lesson. Recall that Dylan is trying to make a canvas that has the dimensions of a golden rectangle.
wℓ=21+5
If he cuts the length ℓ of the canvas to be 60 inches. What should be the width w? Round the answer to two decimal places. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02691em;\">w<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"inches","answer":{"text":["37.08"]}}
Hint
Substitute 60 for ℓ and solve for w.
Solution
The desired width w of the canvas can be found by substituting the known length 60 for ℓ into the equation and solving for w.
The right-hand side of the equation can be simplified by rationalizing the denominator using conjugates.
wℓ=21+5
Substitute
ℓ=60
w60=21+5
▼
Solve for w
MultEqn
LHS⋅2=RHS⋅2
w60(2)=1+5
MoveRightFacToNum
ca⋅b=ca⋅b
w120=1+5
MultEqn
LHS⋅w=RHS⋅w
120=(1+5)w
DivEqn
LHS/(1+5)=RHS/(1+5)
1+5120=w
RearrangeEqn
Rearrange equation
w=1+5120
w=1+5120
▼
Simplify right-hand side
ExpandFrac
ba=b⋅(1−5)a⋅(1−5)
w=(1+5)(1−5)120(1−5)
ExpandDiffSquares
(a+b)(a−b)=a2−b2
w=1−(5)2120(1−5)
PowSqrt
(a)2=a
w=1−5120(1−5)
SubTerm
Subtract term
w=-4120(1−5)
MoveNegNumToFrac
Put minus sign in front of fraction
w=-4120(1−5)
MovePartNumRight
ca⋅b=ca⋅b
w=-4120(1−5)
CalcQuot
Calculate quotient
w=-30(1−5)
UseCalc
Use a calculator
w=37.082039…
RoundDec
Round to 2 decimal place(s)
w≈37.08
The golden ratio is a ratio of two positive numbers such that the ratio of the larger number to the smaller number is equal to the ratio of the sum of the numbers to the larger number.
ϕ=21+5
In the Fibonacci Sequence, consecutive numbers increase according to the golden ratio. Numbers in the sequence can be approximated according to the following equation.
an−1an≈21+5
The 13th term of the Fibonacci Sequence is 233. Use the golden ratio to find the 12th term. Round the answer to the nearest integer.
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The ratio of consecutive terms of the Fibonacci Sequence approximates the golden ratio. With this in mind, we will write a ratio involving the 13^(th) and 12^(th) terms of the sequence. a_(13)/a_(12)=1+sqrt(5)/2 We are told that the 13^(th) term is 233. Knowing this, we can substitute it into the ratio and solve for a_(12). Let's do it!
Since the Fibonacci sequence consists of integers, we round the value down to the nearest integer. This means that the 12^(th) of the sequence is 144.
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Operating with Square Roots
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