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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 th