Algebra/Chapter 1/Exponents and Roots

Operations in Arithmetic

Algebra Chapter 1: Elementary Arithmetic Section 3: Exponents and Roots

Exponents and Roots

1.4: Exponents and Roots

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Exponentiation

What are exponents?

Exponents are a shorthand used for repeated multiplication of the same number. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 × 5 = 5 + 5 + 5 + 5. The expression "4 × " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript (that is, a smaller number written above) after a regular-sized number.

For example: 2³ = 2 × 2 × 2 = 8. The number in larger font is called the base. The number in superscript is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is called the base and 3 is called the exponent.

The expression 2³ is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples:

6 × 6 = 6² (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".)
7 × 7 × 7 × 7 = 7⁴ (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for raised to the fourth power. It is only the second and third powers that usually get abbreviated because they come up more often. When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".)

There are two other powers that play an important role in understanding powers. One is

$2^{0}$

and the other is

$2^{1}$

Any number raised to the power of zero is equal to 1. For example: $6^{0}=1$
Any number raised to the power of one is itself. For example: $15^{1}=15$

In general, an exponent of a number to power of n

a × a × a ... = $a^{n}$

The base is a and it is multiplied by itself n times.

Exponentiation Problems

Roots

What are Roots?

Roots are the opposite for exponents. It's easy, although perhaps long, to compute exponents given a root. For instance 7*7*7*7 = 49*49 = 2401. So, we know the fourth root of 2401 is 7, and the square root of 2401 is 49. What is the third root of 2401? This article gives a formula for determining the answer, while this article gives a detailed explanation of roots.

Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When we graph functions we will see that polynomials that use exponentiation use curves instead of lines. Using algebra we will see that not all of these polynomials are functions, that knowing when a polynomial is a relationship or a function can allow us to make certain types of assumptions, and we can use these assumptions to build mental models for topics that would otherwise be impossible to understand.

For now we will deal with roots by turning them back into exponents.

The positive nth root of $x$ is represented as ${\sqrt[{n}]{x}}=r$. We get rid of the root by raising our answer to the nth power, $r^{n}=x.\!\,$

Problems Using Roots