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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.

Rational Numbers
Fraction
Ratio

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 = \frac{1 + 5}{2}

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.

\frac{ℓ}{w} = 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

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.

2 π = 1.41421356237 \dots = 3.14159265359 \dots

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 - Q or R ∖ 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 \cdot 4 - 4 \cdot (- 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

4

is used. For example, the square root of

16

is denoted as

16 .

16 = 4

In the example above, the principal root of

16

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)
$1$	$1 = 1$	$2$	$2 \approx 1.414213 \dots$
$4$	$4 = 2$	$3$	$3 \approx 1.732050 \dots$
$9$	$9 = 3$	$5$	$5 \approx 2.236067 \dots$
$16$	$16 = 4$	$10$	$10 \approx 3.162277 \dots$
$25$	$25 = 5$	$20$	$20 \approx 4.472135 \dots$

Extra

Square Roots of Fractions and Decimal Numbers

Separate 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.

\frac{9}{1 6} = \frac{3 ^{2}}{4 ^{2}} \Rightarrow \frac{9}{1 6} = \frac{3}{4}

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 = \frac{3 6}{1 0 0} = \frac{6 ^{2}}{1 0 ^{2}} ⇓ \frac{3 6}{1 0 0} = \frac{6}{1 0} = 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!

Grandpa says that there is enough fertilizer to cover

81

square feet. Emily wants to use this fertilzer to make a garden in the shape of a square.

What should be the length

ℓ

of one side of the garden?

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 = ℓ^{2} substitute 81 = ℓ^{2}

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.

ℓ = 81

CalcRoot

Calculate root

ℓ = 9

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.

$a b = a \cdot b,$ for $a \geq 0$ and $b \geq 0$

Proof

Let

x,

y,

and

z

be three non-negative numbers such that

x = a,

y = b,

and

z = a b .

By the definition of a square root, each of these numbers squared is equal to its corresponding radicand.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x^{2} = a y^{2} = b z^{2} = a b (I) (II) (III)

Next, multiply Equation (I) by

y^{2} .

x^{2} = ⇓ x^{2} y^{2} = a a y^{2}

Now, substitute Equations (II) and (III) into this equation.

x^{2} y^{2} = a y^{2}

▼

Substitute values and simplify

Substitute

$y^{2} = b$

x^{2} y^{2} = a b

Substitute

$a b = z^{2}$

x^{2} y^{2} = z^{2}

ProdPowII

$a^{m} b^{m} = (a b)^{m}$

(x y)^{2} = z^{2}

RearrangeEqn

Rearrange equation

z^{2} = (x y)^{2}

Since

x,

y,

and

z

are non-negative, the final equation implies that

z = x y .

z^{2} = (x y)^{2} \Rightarrow z = x y

The last step is substituting

z = a b,

x = a,

and

y = b

into this equation.

z = x y \Leftrightarrow a b = a \cdot 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.

Help Emily and Auntie Agent find the area of the new plot.

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.

A = ℓ w

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
$ℓ^{2} = 160$	$ℓ = 160$
$w^{2} = 360$	$w = 360$

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.

A = ℓ w

SubstituteII

$ℓ = 160$ , $w = 360$

A = 160 \cdot 360

ProdSqrt

$a \cdot b = a \cdot b$

A = 160 \cdot 360

Multiply

A = 57600

CalcRoot

Calculate root

A = 240

Therefore, the area of the new plot is

240

square meters. Auntie Agent is ready to wheel and deal!

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.

The teacher said that the radicand ought to be factored using perfect squares. Then, the Product Property of Square Roots can be used. The teacher suggested to simplify

18

using this method. Help Auntie Agent rewrite

18

in terms of

2 .

Write the exact value, not an approximation.

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 \cdot 2

Knowing this, the Product Property of Square Roots can be used.

18

SplitIntoFactors

Split into factors

9 (2)

SqrtProd

$a \cdot b = a \cdot b$

9 \cdot 2

CalcRoot

Calculate root

32

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 $\frac{a}{b}$ equals the quotient of the square roots of $a$ and $b .$

$\frac{a}{b} = \frac{a}{b},$ for $a \geq 0,$ $b > 0$

Proof

Let

x,

y,

and

z

be non-negative numbers such that

x = a,

y = b,

and

z = \frac{a}{b} .

By the definition of a square root, each of these numbers squared is equal to its corresponding radicand.

⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ x^{2} = a y^{2} = b z^{2} = \frac{a}{b} (I) (II) (III)

Since

b

is a positive number,

y^{2}

is also positive. Therefore, Equation (I) can be divided by

y^{2} .

x^{2} = ⇓ \frac{x ^{2}}{y ^{2}} = a \frac{a}{y ^{2}}

Now, substitute Equations (II) and (III) into this equation.

\frac{x ^{2}}{y ^{2}} = \frac{a}{y ^{2}}

▼

Substitute values and simplify

Substitute

$y^{2} = b$

\frac{x ^{2}}{y ^{2}} = \frac{a}{b}

Substitute

$\frac{a}{b} = z^{2}$

\frac{x ^{2}}{y ^{2}} = z^{2}

$\frac{a ^{m}}{b ^{m}} = (\frac{a}{b})^{m}$

(\frac{x}{y})^{2} = z^{2}

RearrangeEqn

Rearrange equation

z^{2} = (\frac{x}{y})^{2}

Since

x,

y,

and

z

are non-negative, the final equation implies that

z = \frac{x}{y} .

z^{2} = (\frac{x}{y})^{2} \Rightarrow z = \frac{x}{y}

The last step is substituting

z = \frac{a}{b},

x = a,

and

y = b

into this equation.

z = \frac{x}{y} \Leftrightarrow \frac{a}{b} = \frac{a}{b}

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.

Emily's cousin knows that the Pythagorean Theorem can be used to find the hypotenuse

c

of the triangle. After some algebraic manipulation they managed to isolate

c .

a^{2} + b^{2} = c^{2} ⇓ c = a^{2} + b^{2}

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.

Help Emily and their cousin by simplifying the expression for the hypotenuse of the triangle.

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.

c = \frac{2 5}{4 9}

SqrtQuot

$\frac{a}{b} = \frac{a}{b}$

c = \frac{2 5}{4 9}

CalcRoot

Calculate root

c = \frac{5}{7}

Therefore, the length of the hypotenuse is

\frac{5}{7} .

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.

\frac{3}{6}

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.

\frac{3}{6}

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.

Multiply by the Denominator

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 .

\frac{3 \cdot 6}{6 \cdot 6}

Simplify the Fraction

The fraction can now be simplified.

\frac{3 \cdot 6}{6 \cdot 6}

MultSqrt

$a \cdot a = a$

\frac{3 \cdot 6}{6}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{6}{2}

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 + b c

is obtained by switching the sign of the irrational term.

Number a + b c a - b c switch sign switch sign Conjugate a - b c a + b c

For example, the conjugate of

2 - 53

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.

\frac{2}{5 + 3}

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.

Multiply by the Conjugate of the Denominator

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 .

\frac{2 ( 5 - 3 )}{( 5 + 3 ) ( 5 - 3 )}

Simplify the Expression

The obtained expression can now be simplified.

\frac{2 ( 5 - 3 )}{( 5 + 3 ) ( 5 - 3 )}

ExpandDiffSquares

$(a + b) (a - b) = a^{2} - b^{2}$

\frac{2 ( 5 - 3 )}{5 ^{2} - ( 3 ) ^{2}}

CalcPow

Calculate power

\frac{2 ( 5 - 3 )}{2 5 - ( 3 ) ^{2}}

PowSqrt

$(a)^{2} = a$

\frac{2 ( 5 - 3 )}{2 5 - 3}

SubTerm

Subtract term

\frac{2 ( 5 - 3 )}{2 2}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

\frac{5 - 3}{1 1}

The expression has now been rationalized because the denominator is an integer number.

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!

Emily then wants to teach Dylan how to simplify a numeric expression which represents the ratio of the free fall time of two different objects.

\frac{1 + 4}{9 + 2}

What is the simplest way of writing this expression?

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.

\frac{1 + 4}{9 + 2}

CalcRoot

Calculate root

\frac{1 + 2}{3 + 2}

AddTerms

Add terms

\frac{3}{3 + 2}

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.

\frac{3}{3 + 2} \cdot \frac{3 - 2}{3 - 2}

The expression can be now be simplified.

\frac{3}{3 + 2} \cdot \frac{3 - 2}{3 - 2}

MultFrac

Multiply fractions

\frac{3 ( 3 - 2 )}{( 3 + 2 ) ( 3 - 2 )}

Distr

Distribute $3$

\frac{9 - 3 2}{( 3 + 2 ) ( 3 - 2 )}

▼

Simplify denominator

ExpandDiffSquares

$(a + b) (a - b) = a^{2} - b^{2}$

\frac{9 - 3 2}{3 ^{2} - ( 2 ) ^{2}}

CalcPow

Calculate power

\frac{9 - 3 2}{9 - ( 2 ) ^{2}}

PowSqrt

$(a)^{2} = a$

\frac{9 - 3 2}{9 - 2}

SubTerm

Subtract term

\frac{9 - 3 2}{7}

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.

$a n x \pm b n x = (a \pm b) n x$

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

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,

n x

is a real number. This means that the Distributive Property can be used to factor out

n x .

a n x \pm b n x

FactorOut

Factor out $n x$

(a \pm b) n x

If the radicands are not the same but one or both radical terms can be rewritten to have the same radicand, the expression can be simplified by adding or subtracting radicals. Consider the following example.

38 + 22

This expression can be simplified by rewriting

8

in terms of

2 .

38 + 22

▼

Simplify

SplitIntoFactors

Split into factors

3 4 \cdot 2 + 22

SqrtProd

$a \cdot b = a \cdot b$

342 + 22

CalcRoot

Calculate root

3 \cdot 22 + 22

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.

\frac{3 3 + 1 2}{2 4 - 6}

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.

Help Ali and Davontay simplify the given expression!

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.

\frac{3 3 + 1 2}{2 4 - 6}

▼

Simplify numerator

SplitIntoFactors

Split into factors

\frac{3 3 + 4 ( 3 )}{2 4 - 6}

SqrtProd

$a \cdot b = a \cdot b$

\frac{3 3 + 2 3}{2 4 - 6}

AddTerms

Add terms

\frac{5 3}{2 4 - 6}

▼

Simplify denominator

SplitIntoFactors

Split into factors

\frac{5 3}{4 ( 6 ) - 6}

SqrtProd

$a \cdot b = a \cdot b$

\frac{5 3}{2 6 - 6}

SubTerm

Subtract term

\frac{5 3}{6}

Finally, the denominator will be rationalized.

\frac{5 3}{6}

▼

Simplify

MovePartNumLeft

$\frac{a \cdot b}{c} = a \cdot \frac{b}{c}$

5 \frac{3}{6}

DivSqrt

$\frac{a}{b} = \frac{a}{b}$

5 \frac{3}{6}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

5 \frac{1}{2}

SqrtQuot

$\frac{a}{b} = \frac{a}{b}$

5 \frac{1}{2}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

\frac{5}{2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

\frac{5 2}{2 \cdot 2}

MultSqrt

$a \cdot a = a$

\frac{5 2}{2}

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.