.Its value is given by the following expression.
From the examples given above, and are called the square root of the square root of and the square root of respectively.
is used. For example, the square root of is denoted as
|Principal Root of Perfect Squares||Principal Root of Non-Perfect Squares|
|Perfect Square|| Principal Root
|Non-Perfect Square|| Principal 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!
Sometimes it is necessary to simplify a square root. The Product Property of Square Roots can be helpful when doing so.
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.
Since the areas of the square plots are known, it is possible to find and
|Area of Square Plot||Side Length|
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 appears to be twice the value of the square root of Auntie Agent, curious to know why, checks her kid's notes and sees the following notes from his class.
Factor using perfect squares.
When working with square roots, just like how the product of a square root operates, there is a similar property for quotients.
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.
When the denominator of a numeric expression has a number in this form, it can be rationalized by following a standard procedure.
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!
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.
Substitute for and solve for