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Here are a few recommended readings before getting started with this lesson.
φ.Its value is given by the following expression.
Some numbers cannot be expressed as the ratio of two integers. These numbers have a special name.
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.
4is used. For example, the square root of 16 is denoted as
16.
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… |
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 is the square root of 81?
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
y2=b
ab=z2
ambm=(ab)m
Rearrange equation
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.Use the formula for the area of a square and the formula for the area of a rectangle.
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 |
ℓ=160, w=360
a⋅b=a⋅b
Multiply
Calculate root
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.Factor 18 using perfect squares.
Use the Product Property of Square Roots to simplify the given 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
y2=b
ba=z2
bmam=(ba)m
Rearrange equation
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.Use the Quotient Property of Square Roots.
ba=ba