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Roots of Real Numbers
Math 97 Supplement 1
LEARNING OBJECTIVES
- Calculate the exact and approximate value of the square root of a real number.
- Calculate the exact and approximate value of the cube root of a real number.
- Simplify the square root of a real number.
- Apply the Pythagorean Theorem to find the hypotenuse of a right triangle.
The Definition of Square Roots
A square root of a number is a number that when multiplied by itself yields the original number.
For example, 4 is a square root of 16, because 42 16. Since 4 ^2 16 , we can say that −4 is a
square root of 16 as well. Every positive real number has two square roots, one positive and one
negative. For this reason, we use the radical sign to denote the principal (nonnegative)
square root and a negative sign in front of the radical to denote the negative square root.
16 4 Positive square root of 16 16 (^4) Negative square root of 16
Zero is the only real number with exactly one square root.
0 0
If the radicand , the number inside the radical sign, is nonzero and can be factored as the square of another nonzero number, then the square root of the number is apparent. In this case, we have the following property:
a^2 a^ , if a^ ^0
So, if we have 9 , that is the same as 32 , which is the same as 3. It is important to point out that a is required to be nonnegative.
Note that if a 0 , then we would get something like this:
2 ^3 ^9 ^3 , and not -3.
Squaring the negative will make it positive, and the square root will keep it positive.
Some will make the definition that^ a^2^ a since both sides keep the sign the same when a is
positive, but changes the sign of a when a is negative. You will investigate this further in a later course.
Example 1
Find the square root: a. 121 b. 0.
c.
Solution: a.^121 ^112 ^11 b. 0.25 0.5^2 0.
c.
2 4 2 2 9 3 3
Example 2
Find the negative square root: a. 64 b. 1
Solution: a. 64 82 8 b. 1 12 1
The radicand may not always be a perfect square. If a positive integer is not a perfect square,
then its square root will be irrational. Consider 5 , we can obtain an approximation by bounding it using the perfect squares 4 and 9 as follows:
With this we conclude that 5 is somewhere between 2 and 3. This number is better
approximated on most calculators using the square root button,.
Note: Use [2nd][ x^2 ] on the TI-83 or TI-84.
5 2.236 because 2.236^2 5. Note: Using more decimals gives more accurate results.
Example 3
Find the cube root: a.^3 b.^3
c.^3
Solution: a.^3 125 3 53 5 b.^3 0 3 03 0
c.
3 3 8 3 2 2 27 3 3
Example 4
Find the cube root: a.^3 27 b.^3 1
Solution: a. 3 3 3 27 3 3
b. 3 3 3 1 1 1
It may be the case that the radicand is not a perfect cube. If this is the case, then its cube root will
be irrational. For example, 3 2 is an irrational number, which can be approximated on most
scientific calculators using the root button x or by using a power of (^1) x.
On the TI83-84, you can hit [Math], then use option 4 for the cube root. You can also you’re your number followed by ^(1/3) to compute a cube root.
Therefore, we have 3 2 1.260, because 1.260^3 2.
Note: To calculate n^2 with other indices, you can use 2^(1/n) on the calculator to compute higher index roots. It is important to point out that a square root has index 2; therefore, the following are equivalent: (^2) a a
In other words, if no index is given, it is assumed to be the square root.
Simplifying Square Roots
It will not always be the case that the radicand is a perfect square. If not, we use the following
two properties to simplify the expression. Given real numbers A and B where B ≠0,
Product Rule for Radicals : A B A B
Quotient Rule for Radicals :
A A
B B
A simplified radical is one where the radicand does not consist of any factors that can be written as perfect powers of the index. That means a square root, once it is simplified, will not contain any factors that are perfect squares still inside the root. A cube root, once it is simplified, will not contain any factors that are perfect cubes still inside the root. Given a square root, the idea is to identify the largest square factor of the radicand and then
apply the property shown above. As an example, to simplify 12 , notice that 12 is not a perfect square. However, 12 does have a perfect square factor of 4, 12 = 4⋅3. Apply the property as follows:
12 4 3 Apply the product rule for radicals 4 (^3) Simplify the square root of the perfect square 2 3
The number 2 3 is a simplified irrational number, and is considered an exact answer. You may be asked to find an approximate answer rounded off to a certain decimal places at the end of a problem. In that case, use a calculator to find the decimal approximation using either the non- simplified root or the simplified equivalent.
As a check, calculate 12 and 2 3 on a calculator and verify that the results are both approximately 3.46.
List of Commonly Used Perfect Powers
Squares: 22 4 32 9 42 16 52 25 62 36 7 2 49
Cubes: 23 8 33 27 43 64 53 125 63 216 73 343
4 th^ Powers: 24 16 34 81 44 256 54 625
5 th^ Powers: 25 32 35 243 45 1024
Example 7 shows how you can write your work when you recognize perfect squares like those given in the “List of Commonly Used Perfect Powers” on page 5. If you recognize that 81 is the
same as 9^2 , then when you write 81 in your work, you can simplify it to 9 in the next step.
Example 7
Simplify:5 162
Solution:
Answer: 45 2
Try this!
Simplify: 4 150.
Answer: 20 6
Roots and Exponents – Inverse Operations
One of the main ideas to see with exponents and roots is that they are inverse operations. Consider the following calculations:
2 2
When you take the square root of a positive number squared, you get the same number. When you square the square root of a positive number, you get the same number. Notice that it does not matter if we apply the exponent first or the square root first. This is true for any positive real number. We can then say the following for square roots:
2 2 a a a when a 0
Because the square and the square root “undo each other”, we can say that they are inverse operations for positive real numbers.
Notice the same thing is true for cubes and cube roots:
3 3
3 3
and
3 3
3 3
In this example, the cube and cube root are inverse operations, and we can make the following statement about cube roots:
3 3 3 3 a a a
Notice that with cube roots, this works for negative numbers as well since the cube roots of negative numbers are still real numbers.
Example 10
Simplify: 3 3 10
Solution:
Apply the fact that 3 3 a a.
3 3 10 10
Pythagorean Theorem
There is another way to think about square roots, and that is in a geometric context. If you look at the figure shown, the area of the big box is 4, and half of the area is shaded, so the area of the shaded portion is 2. Since it is a square, the
area is given by the formula side side , or 2 side. This
means the length of the sides of the shaded region is 2 ,
since (^) 2 2 Area side 2 2.
KEY TAKEAWAYS
The square root of a number is a number that when squared results in the original number. The principal square root of a positive real number is the positive square root. The square root of a negative number is currently left undefined. When simplifying the square root of a number, look for perfect square factors of the radicand. Apply the product or quotient rule for radicals and then simplify. The cube root of a number is a number that when cubed results in the original number. Every real number has only one real cube root. When simplifying cube roots, look for perfect cube factors of the radicand. Apply the product or quotient rule for radicals and then simplify. Exponents and roots with a matching index are inverse operations. The Pythagorean theorem gives us a property of right triangles: a^2 b^2^ c^2 and c a^2 b^2 when a and b represent the lengths of the legs of a right triangle and c represents the length of the hypotenuse of the right triangle.
TOPIC EXERCISES
PART A: SQUARE AND CUBE ROOTS
Simplify.
81
49
^16
100
10. ^25
-
2 5
- 2 6
18.^8
19.^3
20.^3
21.^3 ^27
22.^3 ^1
23.^3
24.^3 0.
25.^3.
27.^3
- 3 3 8
- 3 3 15
30.^3
31.^3
32.^3
33.^3
35.^4
38.^8 3 ^8
Use a calculator to approximate to the nearest hundredth.
39.^3
- 10
- 19 42.^7
- 3 5 44. 2 3
45.^3
46.^3
47.^3
48.^3
- Find the squares of the first twelve positive integers.
- Find the cubes of the first twelve positive integers.
PART B: SIMPLIFYING SQUARE ROOTS
Simplify.
53.^18
50
24
40
60.^3
61.^5
64.^5 ^160
65.
2 64
66.
2 25
-
2 2
68.
2 6
PART C: PYTHAGOREAN THEOREM
- If the two legs of a right triangle measure 3 units and 4 units, then find the length of the hypotenuse.
- If the two legs of a right triangle measure 6 units and 8 units, then find the length of the hypotenuse.
ANSWERS
5.^54
7.^12
- Not a real number.
- 5
- 16
- −
- 4
- −
- 0
- −
- −
31.^34
^13
32
24
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Not a real number.
64
2
5 units
7 2 units
3 2 centimeters
2 3 centimeters
4 5 centimeters
7 meters
The diagonal must measure approximately 67.9 inches.
Answers will vary. Either an error in the calculator, or a number times i , depending on the calculator mode.
