Lesson Objectives

- Demonstrate an understanding of how to evaluate exponential expressions
- Learn how to identify the parts of a radical
- Learn how to find the square root of a number
- Learn how to determine if a number is rational, irrational, or not real
- Learn how to find the cube root of a number
- Learn how to find higher level roots

## How to Find the Square Root of a Number

At this point, we should know that addition and subtraction are opposite operations. This means one can be used to undo the other.

12 + 5 = 17

17 - 5 = 12

Similarly, we have multiplication and division. These also represent opposite operations.

12 • 5 = 60

60 ÷ 5 = 12

In this lesson, we will learn how to work with roots (or radicals). When we work with roots, we are undoing an exponential operation. Roots and applying exponents are opposite operations. To begin, let's think about squaring a number, this means we are raising the number to the power of 2. In other words, we are multiplying the number by itself:

What happens when we want to go backward? In other words, suppose we saw the following:

?

What number squared is 100. We need a number that when multiplied by itself gives us 100. Many of you will know immediately that one such number is 10.

10

Therefore, 10 is known as the square root of 100. A square root of a number (100) is a number (10) that multiplied by itself (10 • 10) gives the number (100). Additionally, (-10) is also a square root of 100 since (-10 • -10 = 100).

When we want to ask for the square root of a number, we place the number under a radical symbol: $$\sqrt{100} = 10$$ When we look at a radical, we have a few terms to learn:

Radicand - number under the radical symbol (also called an argument)

Index - number placed at the top left, it is used in a similar manner to an exponent. An index of 2, tells us to find a number that when multiplied by itself produces the radicand. Similarly, an index of 3 tells us to find a number that when multiplied by itself three times produces the radicand.

Radical symbol - this refers to the symbol itself

Radical - refers to the entire expression In our image above, we notice a visible index of 2, when this occurs we have a square root. For a square root, the index does not need to be shown. In other words: $$\sqrt[2]{4} = \sqrt{4}$$ In each case, we are looking for the number that when squared gives us 4. The two answers would be 2 and (-2). Let's talk a bit about notation. When we take the square root of a positive number, we will always have a positive square root and a negative square root. This is due to the sign rules for multiplication:

(-) • (-) = +

(+) • (+) = +

When two negatives or two positives are multiplied, the result is positive. Therefore each positive number will have a positive and a negative square root. When the symbol has no sign in the front, we are looking for the principal or positive square root: $$\sqrt{4} = 2$$ When the symbol has a negative sign in the front, we are looking for the negative square root: $$-\sqrt{4} = (-2)$$ Let's look at a few examples.

Example 1: Find each square root $$\sqrt{900}$$ If we want to find the square root of a number without using a calculator, factor the number and look for pairs:

900 = 2 • 2 • 3 • 3 • 5 • 5

(2 • 3 • 5) • (2 • 3 • 5)

(30)(30)

(30)

Since 30 squared is 900, the principal square root of 900 is 30.

We may run into the square root of a fraction. When this occurs, split the operation and think about the square root of the numerator and denominator separately. Let's look at an example.

Example 2: Find each square root $$\sqrt{\frac{100}{225}}$$ First, let's split the operation: $$\frac{\sqrt{100}}{\sqrt{225}}$$ We can see from our table above, that 10

Example 3: Perform each indicated operation $$\left(\sqrt{17}\right)^2$$ The square root operation cancels with the squaring operation and we are left with the radicand: $$\left(\sqrt{17}\right)^2 = 17$$ Example 4: Perform each indicated operation $$\left(\sqrt{7x + 2}\right)^2$$ The square root operation cancels with the squaring operation and we are left with the radicand: $$\left(\sqrt{7x + 2}\right)^2 = 7x + 2$$ A rational number is one that can be formed from the quotient of two integers. In decimal form, a rational number will terminate or repeat the same pattern forever. Any number whose square root is a rational number is called a "perfect square". An irrational number is one that cannot be written as the quotient of two integers. In decimal form, the number will not terminate or repeat the same pattern. It continues forever with no pattern. The most famous example of this is the number Pi, which is represented with the symbol: $$\pi$$ Pi is normally approximated to 3.14, but after the 4, we have an infinite number of numbers with no pattern. When our root evaluates to an irrational number, we can leave the number in root form for an exact value or give an approximation using a calculator and rounding. Lastly, not every real number has a square root. Suppose we asked for the following: $$\sqrt{-4}$$ What is the square root of negative four? In other words, what number multiplies by itself to give (-4)? We know from our sign rules that:

(+) • (+) = +

(-) • (-) = +

This means we can never square a number and get a negative result. Therefore, there is no such real number that can be squared to produce (-4). We will see later on that another set of numbers can be used, but for now, we say there is "no real solution" or "not a real number". Let's look at a few examples.

Example 5: Determine if each number is rational, irrational, or not real $$-\sqrt{400}$$ This number is rational since (-20)

Therefore, the cube root of 8 is 2.

When we work with higher level odd roots, we can have a negative radicand. Suppose we saw: $$\sqrt[3]{(-8)}$$ We know from the sign rules of multiplication that an odd number of negatives yields a negative:

(-) • (-) • (-) = - $$(-2)^3 = -8$$ $$\sqrt[3]{(-8)} = -2$$ If we have a higher level even root, we can't have a negative radicand. This will result in "not a real number": $$\sqrt[4]{-16}$$ We know that 2

Example 6: Simplify $$\sqrt[3]{125}$$ $$\sqrt[3]{(5)^3} = 5$$ Since 5

Example 7: Simplify $$\sqrt[4]{256}$$ $$\sqrt[4]{(4)^4} = 4$$ Since 4

12 + 5 = 17

17 - 5 = 12

Similarly, we have multiplication and division. These also represent opposite operations.

12 • 5 = 60

60 ÷ 5 = 12

In this lesson, we will learn how to work with roots (or radicals). When we work with roots, we are undoing an exponential operation. Roots and applying exponents are opposite operations. To begin, let's think about squaring a number, this means we are raising the number to the power of 2. In other words, we are multiplying the number by itself:

Squaring 1 » 20 | |||
---|---|---|---|

Number | Squared | Number | Squared |

1 | 1 | 11 | 121 |

2 | 4 | 12 | 144 |

3 | 9 | 13 | 169 |

4 | 16 | 14 | 196 |

5 | 25 | 15 | 225 |

6 | 36 | 16 | 256 |

7 | 49 | 17 | 289 |

8 | 64 | 18 | 324 |

9 | 81 | 19 | 361 |

10 | 100 | 20 | 400 |

?

^{2}= 100What number squared is 100. We need a number that when multiplied by itself gives us 100. Many of you will know immediately that one such number is 10.

10

^{2}= 100Therefore, 10 is known as the square root of 100. A square root of a number (100) is a number (10) that multiplied by itself (10 • 10) gives the number (100). Additionally, (-10) is also a square root of 100 since (-10 • -10 = 100).

When we want to ask for the square root of a number, we place the number under a radical symbol: $$\sqrt{100} = 10$$ When we look at a radical, we have a few terms to learn:

Radicand - number under the radical symbol (also called an argument)

Index - number placed at the top left, it is used in a similar manner to an exponent. An index of 2, tells us to find a number that when multiplied by itself produces the radicand. Similarly, an index of 3 tells us to find a number that when multiplied by itself three times produces the radicand.

Radical symbol - this refers to the symbol itself

Radical - refers to the entire expression In our image above, we notice a visible index of 2, when this occurs we have a square root. For a square root, the index does not need to be shown. In other words: $$\sqrt[2]{4} = \sqrt{4}$$ In each case, we are looking for the number that when squared gives us 4. The two answers would be 2 and (-2). Let's talk a bit about notation. When we take the square root of a positive number, we will always have a positive square root and a negative square root. This is due to the sign rules for multiplication:

(-) • (-) = +

(+) • (+) = +

When two negatives or two positives are multiplied, the result is positive. Therefore each positive number will have a positive and a negative square root. When the symbol has no sign in the front, we are looking for the principal or positive square root: $$\sqrt{4} = 2$$ When the symbol has a negative sign in the front, we are looking for the negative square root: $$-\sqrt{4} = (-2)$$ Let's look at a few examples.

Example 1: Find each square root $$\sqrt{900}$$ If we want to find the square root of a number without using a calculator, factor the number and look for pairs:

900 = 2 • 2 • 3 • 3 • 5 • 5

(2 • 3 • 5) • (2 • 3 • 5)

(30)(30)

(30)

^{2}= 900Since 30 squared is 900, the principal square root of 900 is 30.

We may run into the square root of a fraction. When this occurs, split the operation and think about the square root of the numerator and denominator separately. Let's look at an example.

Example 2: Find each square root $$\sqrt{\frac{100}{225}}$$ First, let's split the operation: $$\frac{\sqrt{100}}{\sqrt{225}}$$ We can see from our table above, that 10

^{2}is 100 and 15^{2}is 225: $$\frac{\sqrt{100}}{\sqrt{225}} = \frac{10}{15} = \frac{2}{3}$$ When we square the square root operation, the exponent and root will cancel. We will just be left with our radicand. Let's look at an example.Example 3: Perform each indicated operation $$\left(\sqrt{17}\right)^2$$ The square root operation cancels with the squaring operation and we are left with the radicand: $$\left(\sqrt{17}\right)^2 = 17$$ Example 4: Perform each indicated operation $$\left(\sqrt{7x + 2}\right)^2$$ The square root operation cancels with the squaring operation and we are left with the radicand: $$\left(\sqrt{7x + 2}\right)^2 = 7x + 2$$ A rational number is one that can be formed from the quotient of two integers. In decimal form, a rational number will terminate or repeat the same pattern forever. Any number whose square root is a rational number is called a "perfect square". An irrational number is one that cannot be written as the quotient of two integers. In decimal form, the number will not terminate or repeat the same pattern. It continues forever with no pattern. The most famous example of this is the number Pi, which is represented with the symbol: $$\pi$$ Pi is normally approximated to 3.14, but after the 4, we have an infinite number of numbers with no pattern. When our root evaluates to an irrational number, we can leave the number in root form for an exact value or give an approximation using a calculator and rounding. Lastly, not every real number has a square root. Suppose we asked for the following: $$\sqrt{-4}$$ What is the square root of negative four? In other words, what number multiplies by itself to give (-4)? We know from our sign rules that:

(+) • (+) = +

(-) • (-) = +

This means we can never square a number and get a negative result. Therefore, there is no such real number that can be squared to produce (-4). We will see later on that another set of numbers can be used, but for now, we say there is "no real solution" or "not a real number". Let's look at a few examples.

Example 5: Determine if each number is rational, irrational, or not real $$-\sqrt{400}$$ This number is rational since (-20)

^{2}is 400 $$\sqrt{-169}$$ This number is not a real number since there is no such number that when squared gives us (-169) $$\sqrt{13}$$ This number is irrational. Our radicand 13 is not a perfect square. This means there is not a rational number that we can square to produce 13.### Cube Root and Higher Level Roots

We are not limited in exponent operations to just squaring a number, therefore, we are not limited to just finding square roots. A cube root is a radical with an index of 3. Let's suppose we saw: $$\sqrt[3]{8}$$ This tells us to find a number that when cubed or multiplied by itself three times gives us 8. We should know that: $$2^3 = 8$$ $$\sqrt[3]{8} = \sqrt[3]{2^3} = 2$$ The index of 3 cancels with the exponent of 3, the two operations undo each other.Therefore, the cube root of 8 is 2.

When we work with higher level odd roots, we can have a negative radicand. Suppose we saw: $$\sqrt[3]{(-8)}$$ We know from the sign rules of multiplication that an odd number of negatives yields a negative:

(-) • (-) • (-) = - $$(-2)^3 = -8$$ $$\sqrt[3]{(-8)} = -2$$ If we have a higher level even root, we can't have a negative radicand. This will result in "not a real number": $$\sqrt[4]{-16}$$ We know that 2

^{4}is 16, however, we can't ever multiply four negative factors and produce a negative result. Therefore, we will state our answer as "not a real number" Let's look at some examples.Example 6: Simplify $$\sqrt[3]{125}$$ $$\sqrt[3]{(5)^3} = 5$$ Since 5

^{3}is 125, the cube root of 125 is 5.Example 7: Simplify $$\sqrt[4]{256}$$ $$\sqrt[4]{(4)^4} = 4$$ Since 4

^{4}is 256, the fourth root of 256 is 4.
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