# Rational Exponents

## Exponents are shorthand for repeated multiplication and can be used to express an n-th root of a number: b is a number x such that xn = b.

#### Key Points

• The number in larger font is called the base. The number in superscript (that is, the smaller number written above) is called the exponent.

• If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal n-th root of b. It is denoted n√b, where √  is the radical symbol; alternatively, it may be written b1/n.

• A power of a positive real number b with a rational exponent m/n in lowest terms satisfies <equation contenteditable="false">${b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}$.

#### Terms

• A real number that can be expressed as the ratio of two integers.

#### Figures

1. ##### Interactive Graph: Exponential graph with different substitute "b" values

Graphs of $y=b^x$ for various bases (b): 10 (red), e (blue), 2 (green), and ½ (purple). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At $x=1$, the y-value equals the base because any number raised to the power of 1 is the number itself. Most notable is $y=(\frac{1}{2})^x$. Why does it look different than the other graphs?

Exponents are a shorthand used for repeated multiplication. 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 after a regular-sized number.

For example: $2^3=2*2*2$. The number in larger font is called the base. The number in superscript (that is, the smaller number written above) is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is the base and 3 is the exponent.The expression 23 is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples: $6*6=6^2$ (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".) $7*7*7*7=7^4$ (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".)

### Rational Exponents

A rational exponent is a rational number that can be used as another way to write roots. An n-th root of a number b is a number x such that xn = b.

If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to $x^n = b$. This solution is called the principal n-th root of b. It is denoted n√b, where √  is the radical symbol; alternatively, it may be written $b^\frac{1}{n}$. For example: $4^\frac{1}{2}=2$, $8^\frac{1}{3}=2$.

When one speaks of the n-th root of a positive real number b, one usually means the principal n-th root.

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

where m is an integer and n is a positive integer.

Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even. $(−27)^\frac{1}{3}=-3$ , $(−27)^\frac{2}{3}=-9$, and $64^\frac{1}{2}$ has two roots 8 and −8. Since there is no real number x such that x2 = −1, the definition of bm/n when b is negative and n is even must use the imaginary unit i.  A power of a positive real number b with a rational exponent m/n in lowest terms satisfies:

• If n is even, then xn = b has two real solutions;
• if b is positive, which are the positive and negative nth roots.
• The equation has no solution in real numbers if b is negative.
• If n is odd, then xn = b has one real solution.
• The solution is positive if b is positive and negative if b is negative.

Examples of exponents graphed can be seen in this figure Figure 1.

#### Key Term Glossary

base
A number raised to the power of an exponent.
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equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
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exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in $x^3$.
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expression
An arrangement of symbols denoting values, operations performed on them, and grouping symbols. E.g. (2x+4)/2
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imaginary
of a number, having no real part; that part of a complex number which is a multiple of the square root of -1.
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imaginary unit
A complex number, usually denoted with i, that is defined as i^2 = -1
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integer
An element of the infinite and numerable set {...,-3,-2,-1,0,1,2,3,...}.
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principal
The money originally invested or loaned, on which basic interest and returns are calculated.
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A root (of a number or quantity).
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rational number
A real number that can be expressed as the ratio of two integers.
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real number
An element of the set of real numbers.  The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers.
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real numbers
The smallest set containing all limits of convergent sequences of rational numbers.
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root
the number which,when  plugged into the equation, will produce a zero.
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sign
positive or negative polarity.
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term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.