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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 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

Some numbers cannot be expressed as the ratio of two integers. These numbers have a special name.

Concept

The set of irrational numbers is formed by all numbers that *cannot* be expressed as the ratio between two integers.
*not* repeating and non-terminating.

$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 $2 π =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

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 $16.$ *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

### Extra

Square Roots of Fractions and Decimal Numbers

$4⋅4-4⋅(-4) =16=16 $

All positive numbers have two square roots — one positive and one negative. To avoid ambiguity, when talking about $4 $is used. For example, the square root of $16$ is denoted as

$16 .$

$16 =4 $

In the example above, the principal root of $16$ 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) |

$1$ | $1 =1$ | $2$ | $2 ≈1.414213…$ |

$4$ | $4 =2$ | $3$ | $3 ≈1.732050…$ |

$9$ | $9 =3$ | $5$ | $5 ≈2.236067…$ |

$16$ | $16 =4$ | $10$ | $10 ≈3.162277…$ |

$25$ | $25 =5$ | $20$ | $20 ≈4.472135…$ |

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.

$169 =4_{2}3_{2} ⇒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 =10_{2}6_{2} ⇓10036 =106 =0.6 $

Example

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?

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What is the square root of $81?$

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.
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

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 $b≥0$

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 $z=xy.$

$⎩⎪⎪⎨⎪⎪⎧ x_{2}=ay_{2}=bz_{2}=ab (I)(II)(III) $

Next, multiply Equation (I) by $y_{2}.$
$x_{2}=⇓x_{2}y_{2}= aay_{2} $

Now, substitute Equations (II) and (III) into this equation.
$x_{2}y_{2}=ay_{2}$

▼

Substitute values and simplify

Substitute

$y_{2}=b$

$x_{2}y_{2}=ab$

Substitute

$ab=z_{2}$

$x_{2}y_{2}=z_{2}$

ProdPowII

$a_{m}b_{m}=(ab)_{m}$

$(xy)_{2}=z_{2}$

RearrangeEqn

Rearrange equation

$z_{2}=(xy)_{2}$

$z_{2}=(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

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"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:0.68333em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">A<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord Roboto-Regular\">m<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["240"]}}

Use the formula for the area of a square and the formula for the area of a rectangle.

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 |
---|---|

$ℓ_{2}=160$ | $ℓ=160 $ |

$w_{2}=360$ | $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

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"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:1.04em;vertical-align:-0.13278em;\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.90722em;\"><span class=\"svg-align\" style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\" style=\"padding-left:0.833em;\"><span class=\"mord\">1<\/span><span class=\"mord\">8<\/span><\/span><\/span><span style=\"top:-2.86722em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z'\/><\/svg><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.13278em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["3\\sqrt{2}","3\\sqrt[2]{2}"]}}

Factor $18$ using perfect squares.

In order to use the Product Property of Square Roots, the radicand should be factored using perfect squares.
Therefore, $18 $ equals $32 .$ Auntie Agent feels relieved to have figured out what the graphing calculator expressed.

$18=9⋅2 $

Knowing this, the Product Property of Square Roots can be used.
$18 $

SplitIntoFactors

Split into factors

$9(2) $

SqrtProd

$a⋅b =a ⋅b $

$9 ⋅2 $

CalcRoot

Calculate root

$32 $

Pop Quiz

Use the Product Property of Square Roots to simplify the given square roots.

Discussion

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 $b.$

$ba =b a ,$ for $a≥0,$ $b>0$

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 $z=yx .$

$⎩⎪⎪⎪⎨⎪⎪⎪⎧ x_{2}=ay_{2}=bz_{2}=ba (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}=⇓y_{2}x_{2} = ay_{2}a $

Now, substitute Equations (II) and (III) into this equation.
$y_{2}x_{2} =y_{2}a $

▼

Substitute values and simplify

Substitute

$y_{2}=b$

$y_{2}x_{2} =ba $

Substitute

$ba =z_{2}$

$y_{2}x_{2} =z_{2}$

$b_{m}a_{m} =(ba )_{m}$

$(yx )_{2}=z_{2}$

RearrangeEqn

Rearrange equation

$z_{2}=(yx )_{2}$

$z_{2}=(yx )_{2}⇒z=yx $

The last step is substituting $z=ba ,$ $x=a ,$ and $y=b $ into this equation.
$z=yx ⇔ba =b a $

Example

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. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"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\">c<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["5\/7"]}}

Use the Quotient Property of Square Roots.

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

Use the Quotient Property of Square Roots to simplify the given square root.

Discussion

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.

$6 3 $

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.
*expand_more*
*expand_more*

$6 3 $

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

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 ⋅6 3⋅6 $

2

Simplify the Fraction

The fraction can now be simplified.
The fraction has now been rationalized because the denominator is an integer number.

Pop Quiz

Rationalize the denominator of the given numeric expression.