Squares and Square Roots

The exponent that appears most often on the Math 1 test is 2. Although a² can be read “a to the second,” it is usually read “a squared.” You will see 2 as an exponent in many formulas. For example:

A = s² (the area of a square)
A = πr² (the area of a circle)
a² + b² = c² (the Pythagorean theorem)
x² – y² = (x – y)(x + y) (factoring the difference of two squares)

Numbers that are the squares of integers are called perfect squares. You should recognize at least the squares of the integers from 0 through 15.

Chart with x in the first row and x squared in the second row. The first row has the following after x: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. The second row shows these numbers squared: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

Of course, if you need to evaluate 13², you can use your calculator. However, it is often helpful to recognize these perfect squares. That way, if you see 169, you will immediately think “that is 13².”

Two numbers, 5 and –5, satisfy the equation x² = 25. The positive one, 5, is called the square root of 25 and is denoted by the symbol $\sqrt{25} .$ Clearly, each perfect square has a square root: $\sqrt{0} = 0, \sqrt{25} = 5, \sqrt{100} = 10,$ and $\sqrt{169} = 13$ . However, it is an important fact that every positive number has a square root.

Key Fact A12

For any positive number a, there is a positive number b that satisfies the equation b² = a. That number, b, is called the square root of a and is written $\sqrt{a}$ . So, for any positive number $a, \sqrt{a} \times \sqrt{a} = (\sqrt{a})^{2} = a$ .

Key Fact A13

For any positive numbers a and b:

$\sqrt{a b} = \sqrt{a} \times \sqrt{b} and \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} .$

For Example, $\sqrt{3600} = \sqrt{36} \times \sqrt{100} = 6 \times 10 = 60 and \sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$ .

The expression a³ is often read “a cubed.” Numbers that are the cubes of integers are called perfect cubes. You should memorize the perfect cubes in the following table.

Chart with x in the first row and x cubed in the second row. The first row has the following after x: 0, 1, 2, 3, 4, 5, 6, 10. The second row shows these numbers cubed: 0, 1, 8, 27, 64, 125, 216, 1000.

The only other powers you should recognize immediately are the powers of 2 up to 2¹⁰.

Two charts with x in the first row and 2 to the power of x in the second row. The first row has the following after x: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The second row shows 2 to the power of each number: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

In the same way that we write $b = \sqrt{a}$ to indicate that b² = a:

We write $b = \sqrt[3]{a}$ to indicate that b³ = a and call b the cube root of a.
We write $b = \sqrt[4]{a}$ to indicate that b⁴ = a and call b the fourth root of a.
For any integer n ≥ 2, we write $b = \sqrt[n]{a}$ to indicate that bⁿ = a and call b the nth root of a.

For example:

$\sqrt[3]{125} = 5 because 5^{3} = 125 .$
$\sqrt[3]{- 8} = - 2 because (- 2)^{3} = - 8 .$
$\sqrt[10]{1024} = 2 because 2^{10} = 1024 .$

Note that $\sqrt{- 64}$ is undefined because there is no real number x such that x² = –64. If you enter $\sqrt{- 64}$ on your calculator, you will get an error message. (In Section 2-P, you will read about the imaginary unit i and will review the fact that $\sqrt{- 64}$ is 8i.)

Key Fact A14

For any real number a and integer n ≥ 2:

If n is odd, then $\sqrt[n]{a}$ is the unique real number x that satisfies the equation xⁿ = a.
If n is even and a is positive, then $\sqrt[n]{a}$ is the unique positive number x that satisfies the equation xⁿ = a.

We can now expand our definition of exponents to include fractions.

Key Fact A15

For any positive number b and positive integers n and m with n ≥ 2:

$b^{\frac{1}{n}} = \sqrt[n]{b}$
$b^{\frac{m}{n}} = \sqrt[n]{b^{m}} = (\sqrt[n]{b})^{m}$

For example:

\begin{array}{rcl} 4^{\frac{1}{2}} & = & \sqrt{4} = 2 \\ 27^{\frac{1}{3}} & = & \sqrt[3]{27} = 3 \\ 8^{\frac{2}{3}} & = & (\sqrt[3]{8})^{2} = 2^{2} = 4 \\ 8^{\frac{2}{3}} & = & \sqrt[3]{8^{2}} = \sqrt[3]{64} = 4 \\ 16^{- \frac{1}{4}} & = & \frac{1}{16^{\frac{1}{4}}} = \frac{1}{\sqrt[4]{16}} = \frac{1}{2} \end{array}

The laws of exponents, listed in KEY FACT A11, are equally valid if any of the exponents are fractions. For example:

\begin{array}{rclll} (2^{\frac{1}{2}}) (2^{\frac{1}{3}}) & = & 2^{\frac{1}{2} + \frac{1}{3}} & = & 2^{\frac{5}{6}} \\ {(2^{\frac{3}{2}})}^{6} & = & 2^{(\frac{3}{2}) (6)} & = & 2^{9} \end{array}