
1.6: Exponents and Square Roots

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Learning Objectives

• Interpret exponential notation with positive integer exponents.
• Calculate the $$n$$th power of a real number.
• Calculate the exact and approximate value of the square root of a real number.

Exponential Notation and Positive Integer Exponents

If a number is repeated as a factor numerous times, then we can write the product in a more compact form using exponential notation. For example,

$$5\cdot 5\cdot 5\cdot 5=5^{4}$$

The base is the factor, and the positive integer exponent indicates the number of times the base is repeated as a factor. In the above example, the base is $$5$$ and the exponent is $$4$$. In general, if $$a$$ is the base that is repeated as a factor $$n$$ times, then

Figure 1.6.1

When the exponent is $$2$$, we call the result a square. For example,

$$3^{2}=3\cdot 3=9$$

The number $$3$$ is the base and the integer $$2$$ is the exponent. The notation $$3^{2}$$ can be read two ways: “three squared” or “$$3$$ raised to the second power.” The base can be any real number.

\begin{aligned} (3.2)^{2}&=(3.2)(3.2)=10.24 \\ \left(\frac{3}{5} \right)^{2}b&=\left( \frac{3}{5}\right)\left(\frac{3}{5} \right)=\left(\frac{9}{25} \right) \\ (-7)^{2}&=(-7)(-7)=49 \\ -5^{2}&=-5\cdot 5=-25 \end{aligned}

It is important to study the difference between the ways the last two examples are calculated. In the example $$(−7)^{2}$$, the base is $$−7$$ as indicated by the parentheses. In the example $$−5^{2}$$, the base is $$5$$, not $$−5$$, so only the $$5$$ is squared and the result remains negative. To illustrate this, write

$$-5^{2}=-1\cdot 5^{2}=-1\cdot 5\cdot 5=-25$$

This subtle distinction is very important because it determines the sign of the result.

The textual notation for exponents is usually denoted using the caret $$(^)$$ symbol as follows:

\begin{aligned}8^{2}&=8\wedge 2=8*8=64 \\ -5.1^{2}&=-5.1\wedge 2=-5.1*5.1=-26.01 \end{aligned}

The square of an integer is called a perfect square. The ability to recognize perfect squares is useful in our study of algebra. The squares of the integers from $$1$$ to $$15$$ should be memorized. A partial list of perfect squares follows:

$$\{0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,...\}$$

Exercise $$\PageIndex{1}$$

Simplify

$$(−12)^{2}$$.

$$144$$

When the exponent is $$3$$ we call the result a cube. For example,

$$3^{3}=3\cdot 3\cdot 3=27$$

The notation $$3^{3}$$ can be read two ways: “three cubed” or “$$3$$ raised to the third power.” As before, the base can be any real number.

\begin{aligned} \left(\frac{2}{5} \right)^{3}&= \left(\frac{2}{5} \right)\left(\frac{2}{5} \right)\left(\frac{2}{5} \right)=\frac{8}{125} \\ (-7)^{3}&=(-7)(-7)(-7)=-343 \\ -4^{3}&=-4\cdot 4\cdot 4 =-64 \end{aligned}

Note that the result of cubing a negative number is negative. The cube of an integer is called a perfect cube. The ability to recognize perfect cubes is useful in our study of algebra. The cubes of the integers from $$1$$ to $$10$$ should be memorized. A partial list of perfect cubes follows:

$$\{0,1,8,27,64,125,216,343,512,729,1000,...\}$$

Exercise $$\PageIndex{2}$$

Simplify $$(−2)^{3}$$.

$$-8$$

If the exponent is greater than $$3$$, then the notation an is read “a raised to the $$n$$th power.”

\begin{aligned} 10^{6}&=10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10=1,000,000 \\ (-1)^{4}&=(-1)(-1)(-1)(-1)=1 \\ \left(\frac{1}{3} \right)^{5}&=\frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot \frac{1}{3} =\frac{1}{243} \end{aligned}

Notice that the result of a negative base with an even exponent is positive. The result of a negative base with an odd exponent is negative. These facts are often confused when negative numbers are involved. Study the following four examples carefully:

The base is $$(-2)$$ The base is $$2$$
$$\begin{array}{c}{(-2)^{4}=(-2)\cdot (-2)\cdot (-2)\cdot (-2)=+16} \\ {(-2)^{3}=(-2)\cdot (-2)\cdot (-2)=-8} \end{array}$$ $$\begin{array}{c}{-2^{4}=-2\cdot 2\cdot 2\cdot 2=-16}\\{-2^{3}=-2\cdot 2\cdot 2=-8} \end{array}$$

Table 1.6.1

The parentheses indicate that the negative number is to be used as the base.

Example $$\PageIndex{1}$$

Calculate:

1. $$\left(-\frac{1}{3} \right)^{3}$$
2. $$\left(-\frac{1}{3} \right)^{4}$$

Solution:

The base is $$−\frac{1}{3}$$ for both problems.

a. Use the base as a factor three times.

\begin{aligned} \left(-\frac{1}{3} \right)^{3}&=\left(-\frac{1}{3} \right)\left(-\frac{1}{3} \right)\left(-\frac{1}{3} \right) \\ &=-\frac{1}{27} \end{aligned}

b. Use the base as a factor four times.

\begin{aligned} \left(-\frac{1}{3} \right)^{4}&=\left(-\frac{1}{3} \right)\left(-\frac{1}{3} \right)\left(-\frac{1}{3} \right)\left(-\frac{1}{3} \right) \\ &=+\frac{1}{81} \end{aligned}

a. $$-\frac{1}{27}$$; b. $$\frac{1}{81}$$

Exercise $$\PageIndex{3}$$

Simplify:

$$−10^{4}$$ and $$(−10)^{4}$$.

$$−10,000$$ and $$10,000$$

Square Root of a Real Number

Think of finding the square root of a number as the inverse of squaring a number. In other words, to determine the square root of $$25$$ the question is, “What number squared equals $$25$$?” Actually, there are two answers to this question, $$5$$ and $$−5$$.

$$5^{2}=25\quad\text{and}(-5)^{2}=25$$

When asked for the square root of a number, we implicitly mean the principal (nonnegative) square root. Therefore we have,

$$\sqrt{a^{2}}=a$$, if $$a\geq 0$$ or more generally $$\sqrt{a^{2}}=|a|$$

As an example, $$\sqrt{25}=5$$, which is read “square root of $$25$$ equals $$5$$.” The symbol $$√$$ is called the radical sign and $$25$$ is called the radicand. The alternative textual notation for square roots follows:

$$\sqrt{16}=text{sqrt}(16)=4$$

It is also worthwhile to note that

$$\sqrt{1}=1\quad\text{and}\quad\sqrt{0}=0$$

This is the case because $$1^{2}=1$$ and $$0^{2}=0$$.

Example $$\PageIndex{2}$$

Simplify:

$$\sqrt{10,000}$$.

Solution:

$$10,000$$ is a perfect square because $$100⋅100=10,000$$.

\begin{aligned} \sqrt{10,000}&=\sqrt{(100)^{2}} \\ &=100 \end{aligned}

$$100$$

Example $$\PageIndex{3}$$

Simplify:

$$\sqrt{\frac{1}{9}}$$.

Solution:

Here we notice that $$\frac{1}{9}$$ is a square because $$\frac{1}{3}⋅\frac{1}{3}=\frac{1}{9}$$.

\begin{aligned} \sqrt{\frac{1}{9}}&=\sqrt{\left(\frac{1}{3} \right)^{2}} \\ &=\frac{1}{3} \end{aligned}

$$\frac{1}{3}$$

Given $$a$$ and $$b$$ as positive real numbers, use the following property to simplify square roots whose radicands are not squares:

$$\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}$$

The idea is to identify the largest square factor of the radicand and then apply the property shown above. As an example, to simplify $$\sqrt{8}$$ notice that $$8$$ is not a perfect square. However, $$8=4⋅2$$ and thus has a perfect square factor other than $$1$$. Apply the property as follows:

\begin{aligned} \sqrt{8}&=\sqrt{4\cdot 2} \\ &=\color{Cerulean}{\sqrt{4}}\color{black}{\cdot\sqrt{2}} \\ &=\color{Cerulean}{2}\color{black}{\cdot\sqrt{2}}\\&=2\sqrt{2} \end{aligned}

Here $$2\sqrt{2}$$ is a simplified irrational number. You are often asked to find an approximate answer rounded off to a certain decimal place. In that case, use a calculator to find the decimal approximation using either the original problem or the simplified equivalent.

$$\sqrt{8}=2\sqrt{2}\approx 2.83$$

On a calculator, try $$2.83\wedge 2$$. What do you expect? Why is the answer not what you would expect?

It is important to mention that the radicand must be positive. For example, $$\sqrt{−9}$$ is undefined since there is no real number that when squared is negative. Try taking the square root of a negative number on your calculator. What does it say?

Note

Taking the square root of a negative number is defined later in the course.

Example $$\PageIndex{4}$$

Simplify and give an approximate answer rounded to the nearest hundredth:

$$\sqrt{75}$$.

Solution:

The radicand $$75$$ can be factored as $$25 ⋅ 3$$ where the factor $$25$$ is a perfect square.

\begin{aligned} \sqrt{75}&=\sqrt{25\cdot 3}&\color{Cerulean}{The\:largest\:perfect\:square} \\ &=\color{Cerulean}{\sqrt{25}}\color{black}{\cdot\sqrt{3}}&\color{Cerulean}{factor\:of\:75\:is\:25.} \\ &=\color{Cerulean}{5}\color{black}{\cdot\sqrt{3}} \\ &=5\sqrt{3} &\color{Cerulean}{Exact\:answer} \\ &\approx 8.66 &\color{Cerulean}{Approximate\:answer} \end{aligned}

$$\sqrt{75}\approx 8.66$$

As a check, calculate (\sqrt{75}\) and $$5\sqrt{3}$$ on a calculator and verify that the both results are approximately $$8.66$$.

Example $$\PageIndex{5}$$

Simplify:

$$\sqrt{180}$$.

Solution:

\begin{aligned} \sqrt{180}&=\sqrt{36\cdot 5} \\ &=\color{Cerulean}{\sqrt{36}}\color{black}{\cdot\sqrt{5}} \\ &=\color{Cerulean}{6}\color{black}{\cdot\sqrt{5}} \\ &=6\sqrt{5} \end{aligned}

$$6\sqrt{5}$$

Example $$\PageIndex{6}$$

Simplify:

$$-5\sqrt{162}$$.

Solution:

\begin{aligned} -5\sqrt{162}&=-5\cdot\sqrt{81\cdot 2} \\ &=-5\cdot\color{Cerulean}{\sqrt{81}}\color{black}{\cdot\sqrt{2}} \\ &=-5\cdot\color{Cerulean}{9}\color{black}{\cdot\sqrt{2}} \\ &=-45\cdot\sqrt{2} \\ &=-45\sqrt{2} \end{aligned}

$$-45\sqrt{2}$$

Exercise $$\PageIndex{4}$$

Simplify and give an approximate answer rounded to the nearest hundredth:

$$\sqrt{128}$$.

$$8\sqrt{2}≈11.31$$

A right triangle is a triangle where one of the angles measures $$90°$$. The side opposite the right angle is the longest side, called the hypotenuse, and the other two sides are called legs. Numerous real-world applications involve this geometric figure. The Pythagorean theorem states that given any right triangle with legs measuring $$a$$ and $$b$$ units, the square of the measure of the hypotenuse c is equal to the sum of the squares of the measures of the legs: $$a^{2}+b^{2}=c^{2}$$. In other words, the hypotenuse of any right triangle is equal to the square root of the sum of the squares of its legs.

Figure 1.6.1

Example $$\PageIndex{7}$$

If the two legs of a right triangle measure $$3$$ units and $$4$$ units, then find the length of the hypotenuse.

Solution:

Given the lengths of the legs of a right triangle, use the formula $$c=\sqrt{a^{2}+b^{2}}$$ to find the length of the hypotenuse.

Figure 1.6.2

\begin{aligned} c&=\sqrt{a^{2}+b^{2}} \\ c&=\sqrt{3^{2}+4^{2}} \\ &=\sqrt{9+16} \\ &=\sqrt{25} \\ &=5 \end{aligned}

$$c=5$$ units

When finding the hypotenuse of a right triangle using the Pythagorean theorem, the radicand is not always a perfect square.

Example $$\PageIndex{8}$$

If the two legs of a right triangle measure $$2$$ units and $$6$$ units, find the length of the hypotenuse.

Solution:

Figure 1.6.3

\begin{aligned} c&=\sqrt{a^{2}+b^{2}} \\ &=\sqrt{2^{2}+6^{2}} \\ &=\sqrt{4+36} \\ &=\sqrt{40} \\ &=\sqrt{4\cdot 10} \\&=\sqrt{4}\cdot\sqrt{10} \\ &=2\cdot\sqrt{10} \end{aligned}

$$c=2\sqrt{10}$$ units

Key Takeaways

• When using exponential notation $$a^{n}$$, the base $$a$$ is used as a factor $$n$$ times.
• When the exponent is $$2$$, the result is called a square. When the exponent is $$3$$, the result is called a cube.
• Memorize the squares of the integers up to $$15$$ and the cubes of the integers up to $$10$$. They will be used often as you progress in your study of algebra.
• When negative numbers are involved, take care to associate the exponent with the correct base. Parentheses group a negative number raised to some power.
• A negative base raised to an even power is positive.
• A negative base raised to an odd power is negative.
• The square root of a number is a number that when squared results in the original number. The principal square root is the positive square root.
• Simplify a square root by looking for the largest perfect square factor of the radicand. Once a perfect square is found, apply the property $$\sqrt{a⋅b}=\sqrt{a}⋅\sqrt{b}$$, where $$a$$ and $$b$$ are nonnegative, and simplify.
• Check simplified square roots by calculating approximations of the answer using both the original problem and the simplified answer on a calculator to verify that the results are the same.
• Find the length of the hypotenuse of any right triangle given the lengths of the legs using the Pythagorean theorem.

Exercise $$\PageIndex{5}$$ Square of a Number

Simplify.

1. $$10^{2}$$
2. $$12^{2}$$
3. $$(−9)^{2}$$
4. $$−12^{2}$$
5. $$11^{2}$$
6. $$(−20)^{2}$$
7. $$0^{2}$$
8. $$1^{2}$$
9. $$−(−8)^{2}$$
10. $$−(13)^{2}$$
11. $$(\frac{1}{2})^{2}$$
12. $$(−\frac{2}{3})^{2}$$
13. $$0.5^{2}$$
14. $$1.25^{2}$$
15. $$(−2.6)^{2}$$
16. $$−(−5.1)^{2}$$
17. $$(2\frac{1}{3})^{2}$$
18. $$(5\frac{3}{4})^{2}$$

1. $$100$$

3. $$81$$

5. $$121$$

7. $$0$$

9. $$−64$$

11. $$\frac{1}{4}$$

13. $$.25$$

15. $$6.76$$

17. $$5\frac{4}{9}$$

Exercise $$\PageIndex{6}$$ Square of a Number

If $$s$$ is the length of the side of a square, then the area is given by $$A=s^{2}$$.

1. Determine the area of a square given that a side measures $$5$$ inches.
2. Determine the area of a square given that a side measures $$2.3$$ feet.
3. List all the squares of the integers $$0$$ through $$15$$.
4. List all the squares of the integers from $$−15$$ to $$0$$.
5. List the squares of all the rational numbers in the set $$\{0, \frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}, 2\}$$.
6. List the squares of all the rational numbers in the set $$\{0, \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}\}$$.

1. $$25$$ square inches

3. $$\{0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225\}$$

5. $$\{0, \frac{1}{9}, \frac{4}{9}, 1, \frac{16}{9}, \frac{25}{9}, 4\}$$

Exercise $$\PageIndex{7}$$ Integer Exponents

Simplify.

1. $$5^{3}$$
2. $$2^{6}$$
3. $$(−1)^{4}$$
4. $$(−3)^{3}$$
5. $$−1^{4}$$
6. $$(−2)^{4}$$
7. $$−7^{3}$$
8. $$(−7)^{3}$$
9. $$−(−3)^{3}$$
10. $$−(−10)^{4}$$
11. $$(−1)^{20}$$
12. $$(−1)^{21}$$
13. $$(−6)^{3}$$
14. $$−3^{4}$$
15. $$1^{100}$$
16. $$0^{100}$$
17. $$−(\frac{1}{2})^{3}$$
18. $$(\frac{1}{2})^{6}$$
19. $$(\frac{5}{2})^{3}$$
20. $$(−\frac{3}{4})^{4}$$
21. List all the cubes of the integers $$−5$$ through $$5$$.
22. List all the cubes of the integers from $$−10$$ to $$0$$.
23. List all the cubes of the rational numbers in the set $$\{−\frac{2}{3}, −\frac{1}{3}, 0, \frac{1}{3}, \frac{2}{3}\}$$.
24. List all the cubes of the rational numbers in the set $$\{−\frac{3}{7}, −\frac{1}{7}, 0, \frac{1}{7}, \frac{3}{7}\}$$.

1. $$125$$

3. $$1$$

5. $$−1$$

7. $$−343$$

9. $$27$$

11. $$1$$

13. $$−216$$

15. $$1$$

17. $$−\frac{1}{8}$$

19. $$\frac{12}{58}$$

21. $$\{−125, −64, −27, −8, −1, 0, 1, 8, 27, 64, 125\}$$

23. $$\{−\frac{8}{27}, −\frac{1}{27}, 0, \frac{1}{27}, \frac{8}{27}\}$$

Exercise $$\PageIndex{8}$$ Square Root of a Number

Determine the exact answer in simplified form.

1. $$\sqrt{121}$$
2. $$\sqrt{81}$$
3. $$\sqrt{100}$$
4. $$\sqrt{169}$$
5. $$−\sqrt{25}$$
6. $$−\sqrt{144}$$
7. $$\sqrt{12}$$
8. $$\sqrt{27}$$
9. $$\sqrt{45}$$
10. $$\sqrt{50}$$
11. $$\sqrt{98}$$
12. $$\sqrt{2000}$$
13. $$\sqrt{\frac{1}{4}}$$
14. $$\sqrt{\frac{9}{16}}$$
15. $$\sqrt{\frac{5}{9}}$$
16. $$\sqrt{\frac{8}{36}}$$
17. $$\sqrt{0.64}$$
18. $$\sqrt{0.81}$$
19. $$\sqrt{30^{2}}$$
20. $$\sqrt{15^{2}}$$
21. $$\sqrt{(−2)^{2}}$$
22. $$\sqrt{(−5)^{2}}$$
23. $$\sqrt{−9}$$
24. $$\sqrt{−16}$$
25. $$3\sqrt{16}$$
26. $$5\sqrt{18}$$
27. $$−2\sqrt{36}$$
28. $$−3\sqrt{32}$$
29. $$6\sqrt{200}$$
30. $$10\sqrt{27}$$

1. $$11$$

3. $$10$$

5. $$−5$$

7. $$2\sqrt{3}$$

9. $$3\sqrt{5}$$

11. $$7\sqrt{2}$$

13. $$\frac{1}{2}$$

15. $$5\sqrt{3}$$

17. $$0.8$$

19. $$30$$

21. $$2$$

23. Not real

25. $$12$$

27. $$−12$$

29. $$60\sqrt{2}$$

Exercise $$\PageIndex{9}$$ Square Root of a Number

Approximate the following to the nearest hundredth.

1. $$\sqrt{2}$$
2. $$\sqrt{3}$$
3. $$\sqrt{10}$$
4. $$\sqrt{15}$$
5. $$2\sqrt{3}$$
6. $$5\sqrt{2}$$
7. $$−6\sqrt{5}$$
8. $$-4\sqrt{6}$$
9. $$\sqrt{79}$$
10. $$\sqrt{54}$$
11. $$−\sqrt{162}$$
12. $$−\sqrt{86}$$
13. If the two legs of a right triangle measure $$6$$ units and $$8$$ units, then find the length of the hypotenuse.
14. If the two legs of a right triangle measure $$5$$ units and $$12$$ units, then find the length of the hypotenuse.
15. If the two legs of a right triangle measure $$9$$ units and $$12$$ units, then find the length of the hypotenuse.
16. If the two legs of a right triangle measure $$\frac{3}{2}$$ units and $$2$$ units, then find the length of the hypotenuse.
17. If the two legs of a right triangle both measure $$1$$ unit, then find the length of the hypotenuse.
18. If the two legs of a right triangle measure $$1$$ unit and $$5$$ units, then find the length of the hypotenuse.
19. If the two legs of a right triangle measure $$2$$ units and $$4$$ units, then find the length of the hypotenuse.
20. If the two legs of a right triangle measure $$3$$ units and $$9$$ units, then find the length of the hypotenuse.

1. $$1.41$$

3. $$3.16$$

5. $$3.46$$

7. $$−13.42$$

9. $$8.89$$

11. $$−12.73$$

13. $$10$$ units

15. $$15$$ units

17. $$\sqrt{2}$$ units

19. $$2\sqrt{5}$$ units

Exercise $$\PageIndex{10}$$ Discussion Board Topics

1. Why is the result of an exponent of $$2$$ called a square? Why is the result of an exponent of $$3$$ called a cube?
2. Research and discuss the history of the Pythagorean theorem.
3. Research and discuss the history of the square root.
4. Discuss the importance of the principal square root.