8. Square root
Square root
Example 1
a) Calculate the length of one side of a square floor when the floor area is [[$ 25 $]] m [[$ ^ 2 $]].
Solution:
The area of the square is [[$ 25 $]] m [[$ ^ 2 $]], so the length of one side must be [[$ 5 $]] m. This is the case because [[$ 5 $]] m · [[$ 5 $]] m [[$ = 25 $]] m [[$ ^ 2 $]].
b) What if the area of the square floor is [[$ 20 $]] m [[$ ^ 2 $]]. What is the length of one side then?
Solution:
If the side length was [[$ 4 $]] m, the floor area would be [[$ 4 $]] m · [[$ 4 $]] m [[$ = 16 $]] m [[$ ^ 2 $]]. The side length is now not an integer, but must be between [[$ 4 $]] m and [[$ 5 $]] m.
The problem could be solved by experimentation, but the solution is easier to find by taking the square root of [[$ 20 $]] with a calculator.
![radical sign, radicand and square root](https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/8-neli%C3%B6juuri/neli%C3%B6juuri/s2k11:file/photo/eef3ebaf353562e38b075f29771dbaa013eeb89d/8_square-root-example1c.png)
The marking is read as "the square root of 20".
The value of the square root can be calculated by typing the radical sign and 20 to a calculator.
The answer can be checked as follows: ([[$ 4.47 $]]m)[[$ ^2 $]] [[$ = 19.9809 $]] m[[$ ^2 ≈ 20 $]] m[[$ ^2 $]].
Square root
The square root of number [[$ a $]], [[$ \sqrt{a} $]], is the positive number that produces number [[$ a $]] when it is multiplied with itself, or squared.
In other words, [[$ \sqrt{a} = b $]] If [[$ b ^ 2 = a $]] and [[$ b ≥ 0 $]].
Taking the square root of a number is the inverse function of raising the number to a square. The square root thus answers the question, “Which number must be raised to another power to obtain the number in question?”. A square root cannot be taken from a negative number, and the value of a square root is never negative.
Example 2
a) The square root of number [[$ 16 $]] is [[$ \sqrt{16} = 4 $]], because the square of [[$ 4 $]] is [[$ 4^2=16 $]] .
b) The square root of the number [[$ 9 $]] [[$ \sqrt{9} = 3 $]], because the square of the number [[$ 3 $]] is [[$ 3 ^ 2 = 9 $]] .
Example 3
![a-c) Raising to a square and taking a square root cancel one another out. d) Not applicable, because the square root of a negative number cannot be calculated.](https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/8-neli%C3%B6juuri/neli%C3%B6juuri/s2k13:file/photo/c7310d48e112369724cfaf76c5f42e1b26ecd627/8_square-root-example3.png)
Exercises
Basic exercises
Applied exercises
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3/08. Submission folder for answers
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