Exponents

Welcome to our website where you can learn everything about the mathematical operation known as exponentiation. First, we are going to define the base b and the exponent n, also called index or power, and then, along with examples, go on with the exponent rules for the addition, subtraction, division, multiplication as well as the power of a power. Finally, we will be discussing rational indices. Read on to learn everything about exponentiations and exponential notation, and make sure to check out our exponent calculator further below on this page.

Exponent Definition

Let n be a positive integer and b a real number, then the exponentiation usually pronounced fully as “b to the power of n” is defined as follows:

bn = y
b to the power of n = $\underbrace{ {\rm b\hspace{3px} \times\hspace{7px} …\hspace{5px} \times\hspace{3px} b} }_{\rm n\hspace{3px} times}$

For example, with b = 5 and n = 4, we get 54 = 5 × 5 × 5 × 5 = 625, pronounced in full as “five raised to the fourth power” or, for instance, “five raised to the index of four.”

A shorter form would be “five to the power four”, yet, because not only the word raised is often omitted, but also power and index, 5^4 typically reads as “five to the four” or “five to the fourth”.

Exponent definition: The exponent n of a base number b is a number reduced in size above, to the right of the base, which indicates how many times to repeat the base in a multiplication.

When n = 2, we say b squared or b to the square, and if n = 3 we say b cubed or b to the cube. With b = 6, 6 squared is 36 (6^2), and 6 cubed equals 216 (6^3).

Note that only n=-1,2, and 3 have their own terminologies. When n equals -1 we say reciprocal, a special case of a negative index as discussed in the next section.

Negative Exponents

A negative index of b ≠ 0 means the reciprocal, 1 divided by n times b; in particular, b-1 = 1 / b:

b-n = 1/bn
For example: 2-2 = 1/22 = 1/4. Calculate a negative index by getting the positive power, then take the reciprocal. Swapping the sign of the exponent gives the reciprocal.

Observe that the reciprocal and the inverse of bn, are not the same. As the variable is the base b, the inverse of bn, (bn)−1, is b1/n = $\sqrt[n]{b}$, the nth root of b, futher discussed below.

Note that for the exponential function b^n = y the variable is in the exponent n: b^n = y ⇔ logby = n. (bn)−1 = logby called the logarithm of b to the power of n.

In the next part of our article we are going to discuss integer representation.

Exponential Form

Integers can be expressed in various ways, including the standard form, scientific notation, factor form and in exponential form for example, which is synonym for exponential notation.

The number five hundred and twelve in standard form is 512, in product form it can be written as, for example, 4 x 128, and using power notation it is 2^9. The scientific notation is 5.12e+3.

Put simply, exponential form means displaying a number y as bn; this notation is a method for writing really big and really small numbers in simple manner, e.g. 10^10 instead of 10000000000.

Next we have a look at the laws of exponents for the sum, difference, product and quotient of numbers in exponential form, along with the powers of zero as well as base 0 powers.

Exponent Rules

For $b,c\hspace{3px} \varepsilon\hspace{3px} \mathbb{R}$ and $m,n \hspace{3px} \varepsilon\hspace{3px} \mathbb{N^{+}}$ the rules of exponents, also known as laws of exponent, exponential rules and the like are given below, right after the definition of the power 0.

Definition of power of 0: b0 = 1. Furthermore, b1 = b and 0n = 0 (n > 0).

These rules are meant for simplifying exponents, and for each exponent rule, we are going to state the rule, followed by an example, highlighting special cases, in case there are any.

Adding Exponents

bm+n = bm × bn
The number b to the sum m+n of the powers equals bm multiplied by bn. This law is known as sum of powers and explains how to add exponents.
For example: 22+3 = 22 × 23 = 4 × 8 = 32 = 25
Special case: b × bn = bn+1

Subtracting Exponents

bn-m = bn / bm,b ≠ 0
The number b to the difference n-m of the powers equals bn divided by mn. This law is known as difference of powers and explains how to subtract indices.
36-4 = 36 / 34 = 729 / 81 = 9 = 32

Multiplying Exponents

bn × m = (bn)m
The number b to the product n × m of the powers equals (b to the power of n) to the power of m. This law is known as multiplication of powers and explains how to multiply exponents.
42 × 3 = (42)3 = 163 = 4096 = 46

Multiplying Exponents with Different Bases

bm × cm = (bc)m
The multiplication of b to the power of m by c to the power of m equals the product (b × c) to the power of m. This rule is known as power of a product, for indices of different bases.
52 × 32 = 25 x 9 = 225 = (5×3)2 = 152

Dividing Exponents

bn / cn = (b/c)n, c ≠ 0
The numbers b to the power of n divided by c to the power of n is equal to the quotient of b and c to the power of n. This law is known as power of quotient, ruling the division of quotients.
42 / 22 = 16 / 4 = 4 = (4/2)2 = 22

Make sure to understand that bn ≠ nn, unless n=b. That is, exponentiation is not commutative. And it isn’t associative either. In the next paragraph you can find our calculator.

Exponent Calculator

Our calculator is straightforward: Enter the base using decimal notation, the insert the power (which ought to be a decimal, too.) After that, press the convert button.


Bookmark us now, and note that frequent numbers on our site include, but are not limited, to:

In the next section we are going to discuss fractional powers, they correspond to the n-th root of a number of the type y = b^m. Of course, exponent laws are there, too.

Rational Exponents

Rational Exponents, also called fractional exponents go beyond elementary algebra by defining $b^{\frac{m}{n}}$ as $\sqrt[n]{b^{m}}$, such that for the nth root of bm, ${\sqrt[n]{b^{m}}}^n$ = $b^{m}$.

If b is a positive real number and n is an even, positive integer, then there are exactly two real solutions such that to x^n = b; the positive solution is called principal nth root of b.

If b is a negative real number, and n is an even, positive integer, then there is no solution in $\mathbb{R}$.

If b is a real number and n is an odd, positive integer, then there is exactly one real solution. The solution is positive if, and only if, b is positive, and negative if, and only if, b is negative.

In particular, for p = 2, and q = 1 we get n = 2, thus $a^{\frac{1}{2}}$ = $\sqrt[2]{a}$, that is the square root, and for q = 1, and p = 3, we obtain the cube root because $a^{\frac{1}{3}}$ = $\sqrt[3]{a}$.

With $a,b\hspace{3px} \varepsilon\hspace{3px} \mathbb{R}$ and $k, m,n \hspace{3px} \varepsilon\hspace{3px} \mathbb{N^{+}}$, the laws of exponents are as follows:

$a^{\frac{m}{n}}$ = $\sqrt[n]{a^{m}}$
e.g.: $a^{\frac{2}{3}}$ = $\sqrt[3]{a^{2}}$

$\sqrt[n]{ab}= \sqrt[n]{a} \sqrt[n]{b}$
Thus: $\sqrt[2]{7\times 8}= \sqrt[2]{7} \sqrt[2]{8}$

$\sqrt[n]{a/b}= \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$, b ≠ 0
For the sake of example: $\sqrt[6]{4/5}= \frac{\sqrt[6]{4}}{\sqrt[6]{5}}$

$\sqrt[n]{a^{m}} = \sqrt[kn]{a^{km}}$
Therefore: $\sqrt[4]{a^{3}} = \sqrt[k\times4]{a^{k\times3}}$

$\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} = \sqrt[m]{\sqrt[n]{a}}$
By way of illustration: $\sqrt[5]{\sqrt[9]{4}} = \sqrt[5\times 9]{4} = \sqrt[9]{\sqrt[5]{4}}$

Special cases:

$\sqrt[2]{a} = \sqrt{a}$
As an example: $\sqrt[2]{2} = \sqrt{2}$

$\sqrt[1]{a} = a$
Exempli gratia: $\sqrt[1]{3} = 3$

$\sqrt[n]{0} = 0$
To illustrate: $\sqrt[2]{0} = 0$

$\sqrt[n]{a^{-m}} = \frac{1}{\sqrt[n]{a^{m}}}$
So, $\sqrt[n]{4^{-2}} = \frac{1}{\sqrt[n]{4^{2}}}$

In the next paragraph we review the concept of index or power of a number.

What is an Exponent?

Above, we have defined both, exponentiation as well as exponent. In math, the exponent n is a rational number also called index or power, usually written in superscript or with a caret as b^n.

  • If the index n is 0, then bn represents the number one.
  • If the power n is an positive integer, then bn stands for the product n times b.
  • If the power m / n is a fraction of two integers different from zero, then bm/n denotes the nth (principal) root of a number bm, x = $\sqrt[n]{b^{m}}$, such that xn = bm.

Frequent bases include, among others, powers of ten such as 103 for 1000 and 10-2 for 0.001, as well as powers of two, like 25 = 32 and 210 = 1024 for example.

Whereas powers of two, aka binary system, are heavily used in computing, powers of ten correspond to the decimal system, in which the nth power is written as one followed by n 0’s.

BTW: You can locate many numbers in exponential notation using the form located in the sidebar for visitors using a desktop computer, and at the end of this article for users with a mobile device.

Upon entering your search term, you will be taken to a result page including all relevant posts. Give it a try now looking x to the power of ten up; x means a base number of your choosing.

Ahead are the FAQs about the subject matter, as well as the summary of our content.

Exponentiation

You have reached the final part of our article, which we summarize using an image:

Exponentiation
If you have been looking for the properties of exponents or exponential calculator, fractional indices, or power calculator, then you have also found what you have been searching for.

In the context of our site, the frequently asked questions about our topic include, for instance:

  • What is standard notation?
  • What does exponent mean?
  • What are exponents?
  • What is exponent?
  • What is a exponent?
  • What are exponents in math?
  • What is an exponent in math?
  • What are the laws of exponents?

Taking into account our information, you should be in the position to answers them all. Yet, if there is something which needs clarification, drop us a message in the form at the end of this page.

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