Exponent calculator

This exponent calculator computes the result of raising a given base to a specified power. It offers flexibility by allowing you to either input your own base and power values or select “e” as the base with the option to customize the default power value. You can easily switch between using a custom base and power or “e” as the base, and this exponent calculator provides real-time updates of the result as the input values change.

Use "e" as base

Base	Power	Result
5	-	-

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What is an exponent?

An exponent, often referred to as a “power” or “index,” is a mathematical operation that indicates the number of times a base number should be multiplied by itself. It is written as a small, raised number to the right of the base number. The exponent tells you how many times the base number should be multiplied by itself.

Here’s the general form:

a^b

In this formula:

$a$ is the base number.
$b$ is the exponent, which specifies the number of times $a$ is multiplied by itself.

For example, if ( $a = 2$ ) and ( $b = 3$ ), then $\displaystyle 2^3$ means you should multiply 2 by itself three times:

2^3 = 2 \times 2 \times 2 = 8

So, $\displaystyle 2^3$ equals 8.

Exponents are fundamental in mathematics and are used in various mathematical operations, such as exponentiation, logarithms, and scientific notation, to represent and manipulate numbers.

Exponent rules

Product of powers rule:

\displaystyle a^m \times a^n = a^{m+n}

This rule states that when you multiply two powers with the same base, you can add their exponents.

Quotient of powers rule:

\frac{a^m}{a^n} = a^{m - n}

This rule states that when you divide two powers with the same base, you can subtract the exponent of the denominator from the exponent of the numerator.

Power of a power rule:

(a^m)^n = a^{m \times n}

This rule states that when you raise a power to another power, you can multiply the exponents.

Power of a product rule:

(a \times b)^n = a^n \times b^n

This rule states that when you raise a product to a power, you can distribute the power to each factor within the parentheses.

Power of a quotient rule:

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

This rule states that when you raise a quotient to a power, you can raise the numerator and denominator to that power separately.

Zero exponent rule:

a^0 = 1

Any non-zero number raised to the power of 0 is equal to 1.

Negative exponent rule:

a^{-n} = \frac{1}{a^n}

This rule states that a negative exponent is equivalent to taking the reciprocal of the base raised to the positive exponent.

Exponent of 1 rule:

a^1 = a

Any number raised to the power of 1 is equal to itself.

Exponent of -1 rule (reciprocal rule):

a^{-1} = \frac{1}{a}

This rule states that any number raised to the power of -1 is equal to its reciprocal.

These exponent laws and rules are fundamental in algebra and are used to simplify and manipulate expressions involving exponents.

Other key points and information

Here are some additional key points and information about exponents:

The exponent with a base of $e$ : This refers to the natural exponential function, which is denoted as $e^{x}$ . The number $e$ is approximately equal to 2.71828 and is known as Euler’s number. The natural exponential function $e^{x}$ is a fundamental mathematical function used in various areas of mathematics, science, and engineering. It has many important properties and applications, including modeling exponential growth and decay, calculating compound interest, and solving differential equations.
Fractional exponents: Exponents are not limited to whole numbers; they can be fractions or decimals as well. For example, $a^{1/2}$ represents the square root of $a$ , and $a^{1/n}$ represents the $n$ th root of $a$ .
Rational exponents: Rational exponents can be expressed as a combination of roots and powers. For example, $a^{3/2}$ is the same as $\sqrt{a^3}$ , which means taking the square root of $a^{3}$ .
Negative exponents: As mentioned earlier, negative exponents represent taking the reciprocal of the base raised to the positive exponent. For example, $a^{-3}$ is equivalent to $\frac{1}{a^{3}}$ .
Zero exponent: Any nonzero number raised to the power of 0 is equal to 1, as shown by the rule $a^{0}=1$ .
Exponential growth and decay: Exponents are used to model exponential growth and decay in various fields, including finance, biology, and physics. The general form of an exponential growth or decay function is $A(t)=A_0 \times e^{rt}$ , where $A(t)$ is the final amount, $A_0$ is the initial amount, $r$ is the growth or decay rate, and $t$ is time.
Scientific notation: Exponents are used in scientific notation to express very large or very small numbers more concisely. For example, the speed of light, $299,792,458$ meters per second, can be written as $2.99792458×10^{8}$ m/s in scientific notation.
Logarithms: Logarithms are the inverse operation of exponentiation. They are used to solve equations involving exponents and are valuable in various fields, including mathematics, engineering, and computer science.
Exponents and algebra: Exponents are essential in algebra for simplifying expressions, solving equations, and manipulating equations involving variables. They help in solving polynomial equations and working with algebraic expressions.
Exponential functions: Exponential functions, such as $f(x)=a^{x}$ , where $a$ is a positive constant, play a significant role in calculus and the study of functions. They have unique properties, including rapid growth or decay.
Real-world applications: Exponents are used in various real-world applications, including compound interest calculations in finance, population growth in biology, and signal processing in engineering.

Understanding the properties and rules of exponents is crucial for mastering algebra and calculus and for applying mathematical concepts to solve real-world problems. Exponents are a fundamental building block in mathematics and science, and they have a wide range of practical uses in our everyday lives.