2.2 Radicals

Leanne Thompson

2.2 Radicals

In this lesson, we will begin to work with radicals. A radical is an expression written in the form $a\sqrt[n]{x}$ , where n ∈ N.

$\sqrt{\hphantom{99}}$ is the radical sign,
x is the radicand,
n is the index (if the index is not written, it is equal to 2), and
a is the coefficient.

Practice 1

Fill in the blanks.

a) The radical $\sqrt[3]{15}$ has an index of , a radicand of , and a coefficient of .

b) The radical $\sqrt[4]{5}$ has an index of , a radicand of , and a coefficient of .

Square Roots

To find the square root of a number, you must find a number that multiplied by itself equals the original number. Each positive number will have one positive square root and one negative square root. The symbol $\sqrt{\hphantom{99}}$ denotes a square root. A negative sign should be put in front of the square root sign to indicate the negative square root. Examples of finding a square root can be seen below.

$\sqrt{25}$ = 5 because 5 $\times$ 5 = 25
$\sqrt{25}$ = –5 because –5 $\times$ –5 = 25

Practice 2

Evaluate the following square roots without using a calculator.

a) $\sqrt{16}$

b) $-\sqrt{64}$

c) $2\sqrt{100}$

Cube Roots

To find the cube root of a number, you must find a number that multiplied by itself three times equals the original number. Each number will have one cube root. The symbol $\sqrt[3]{\hphantom{99}}$ denotes a cube root. Examples of finding a cube root can be seen below.

$\sqrt[3]{8}$ = 2 because 2 $\times$ 2 $\times$ 2 = 8

$\sqrt[3]{-8}$ = –2 because –2 $\times$ -2 $\times$ –2 = –8.

Practice 3

Evaluate the following cube roots without using a calculator.

a) $\sqrt[3]{27}$

b) $\sqrt[3]{-125}$

c) $\frac{1}{2}\sqrt[3]{64}$

Roots Greater than Three

Roots greater than three can be evaluated in a similar way to square roots and cube roots. Examples of finding a fourth root can be seen below.

$\sqrt[4]{81}$ = 3 because 3 $\times$ 3 $\times$ 3 $\times$ 3 = 81

$-\sqrt[4]{81}$ = –3 because –3 $\times$ –3 $\times$ –3 $\times$ –3 = 81

Practice 4

Evaluate the following roots without using a calculator.

a) $\sqrt[4]{16}$

b) $\sqrt[5]{-32}$

c) $-3\sqrt[4]{10 000}$

Evaluating Radicals Using a Calculator

To evaluate radicals using a calculator, follow the steps below. These steps will work on TI-83 and TI-84 graphing calculators.

Table 2.2.1
Square Roots	Cube Roots	Roots Greater than Three
Press 2^nd Press x² Input the number under the radical sign Press ENTER	Press MATH Select 4: $\sqrt[3]{\hphantom{99}}$ Press ENTER Input the number under the radical sign Press ENTER	Input the number of the root Press MATH Select 5: $\sqrt[x]{\hphantom{99}}$ Input the number under the radical sign Press ENTER

Practice 5

Evaluate the following roots using a calculator. Round answers to the nearest hundredth where necessary.

a) $\sqrt{256}$

b) $\sqrt[3]{26}$

c) $\sqrt[5]{352}$

d) $-5\sqrt[4]{81}$

Perfect Squares

A perfect square is the number you get when you multiply a rational number by itself. Write the following commonly used perfect squares in the blanks below.

1 $\times$ 1 =	6 $\times$ 6 =
2 $\times$ 2 =	7 $\times$ 7 =
3 $\times$ 3 =	8 $\times$ 8 =
4 $\times$ 4 =	9 $\times$ 9 =
5 $\times$ 5 =	10 $\times$ 10 =

Prime Factorization

A factor is a number that evenly divides another number. A number evenly divides another number if there is no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12 because all these numbers evenly divide 12.

A prime number is a whole number greater than 1 whose only factors are 1 and the number itself. For example, 2 and 5 are prime numbers.

List the first 10 prime numbers below:

A composite number is a whole number that has more than two factors. For example, 9 and 12 are composite numbers.

Prime factorization is a method that is used to find the prime numbers that multiply together to give you an original number.

Table 2.2.2
To find the prime factorization of a composite number:
Step 1: Find two factors whose product is the given number, and use these numbers to create two “branches.” Step 2: If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree. Step 3: If a factor is not prime, write it as the product of two factors and continue the process. Step 4: Write the composite number as the product of all the circled primes.

Example 1

Find the prime factorization of 48.

Table 2.2.3
Steps	Solution
Create a prime factorization tree.
Write the prime factorization out.	48 = 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 3

Practice 6

Find the prime factorization of the following numbers:

a) 36

b) 98

c) 180

Homework

Fill in the blanks.

a) The radical $\sqrt[4]{9}$ has an index of , a radicand of , and a coefficient of .

b) The radical $-\sqrt{8}$ has an index of , a radicand of , and a coefficient of .

c) The radical $-\sqrt[3]{5}$ has an index of , a radicand of , and a coefficient of .
Evaluate the following without using a calculator.

a) $\sqrt{36}$

b) $2\sqrt{144}$

c) $-\sqrt[4]{16}$

d) $-\sqrt{121}$

e) $\sqrt{-49}$

f) $\sqrt[3]{64}$

g) $-5\sqrt[3]{125}$

h) $\sqrt[3]{-1}$

i) $\frac{1}{3}\sqrt[3]{27}$

j) $-5\sqrt[4]{81}$
Evaluate the following using a calculator. Round to the nearest tenth where necessary.

a) $\sqrt{70}$ b) $\sqrt{15}$ c) $\sqrt[3]{80}$ d) $\sqrt[4]{65}$ e) $\sqrt[5]{-98}$

f) $\frac{3}{2}\sqrt[3]{8}$

g) $\sqrt{172}$

h) $\sqrt[3]{-2}$

i) $\frac{4}{5}\sqrt{72}$

j) $\sqrt[4]{-17}$

Practice writing the first 10 perfect squares by filling in the blanks.

1 $\times$ 1 =	6 $\times$ 6 =
2 $\times$ 2 =	7 $\times$ 7 =
3 $\times$ 3 =	8 $\times$ 8 =
4 $\times$ 4 =	9 $\times$ 9 =
5 $\times$ 5 =	10 $\times$ 10 =

What is the difference between 9²and $\sqrt{9}$ ?
Why is there no real number equal to $\sqrt{-64}$ ?
Find the prime factorization of the following numbers:

a) 72

b) 60

c) 42

d) 210

e) 450

f) 850

Answers

1.

a) 4, 9, 1

b) 2, 8, –1

c) 3, 5, –1

2.

a) 6	b) 24	c) –2	d) –11	e) undefined
f) 4	g) –25	h) –1	i) 1	j) –15

3.

a) 8.4	b) 3.9	c) 4.3	d) 2.8	e) –2.5
f) 3.0	g) 13.1	h) –1.3	i) 6.8	j) undefined

4. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

5. 9²reads “nine squared” and means nine times itself. The expression $\sqrt{9}$ reads “the square root of nine,” which gives us the number such that if it were multiplied by itself would give you the number inside the square root sign.

6. There is no number that can be multiplied by itself to get –64.

7.

a) 2 × 2 × 2 × 3 × 3	b) 2 × 2 × 3 × 5	c) 2× 3× 7
d) 2× 3× 5× 7	e) 2× 3× 3× 5× 5	f) 2× 5× 5× 17

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Marecek, L., & Honeycutt Mathis, A. (2020). Intermediate algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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