Exponent Solver
An exponent shows how a base is used in a power. For a positive integer exponent, it tells how many times the base is used as a factor. More generally, exponents can represent reciprocals, roots, and variable powers.
Use the Exponent Calculator to evaluate numerical powers, simplify suitable expressions with variables, and work through negative or fractional exponents. It supports Calculator, Math Input, and Canvas input modes, provides step-by-step explanations, and can graph suitable expressions when that option is selected.
How to Solve Exponent Problems
Exponent problems may ask you to evaluate a power, simplify an expression, or solve an equation. First identify the base, the exponent, and the operation connecting the terms.
Evaluating a Power
In \(5^4\), \(5\) is the base and \(4\) is the exponent. Since \(4\) is a positive integer, it means multiply four factors of \(5\):
\[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]
Other numerical powers include:
\[ 3^4 = 81 \]
\[ 5^3 = 125 \]
\[ 8^5 = 32768 \]
\[ 2^8 = 256 \]
A power is the complete expression, such as \(3^4\). The exponent is the raised number, \(4\).
Multiplying and Dividing Powers
When multiplying powers with the same base, add the exponents:
\[ a^m a^n = a^{m+n} \]
\[ x^3x^5 = x^{3+5} = x^8 \]
When dividing powers with the same nonzero base, subtract the exponents:
\[ \frac{a^m}{a^n} = a^{m-n} \]
for \(a \neq 0\).
For example:
\[ \frac{x^8}{x^3} = x^{8-3} = x^5 \]
The condition \(x \neq 0\) is needed because the original expression divides by \(x^3\).
Power of a Power
When a power is raised to another power, multiply the exponents:
\[ (a^m)^n = a^{mn} \]
For example:
\[ (x^2)^5 = x^{2\cdot5} = x^{10} \]
With a coefficient, square both the coefficient and the variable power:
\[ (2x^3)^2 = 2^2(x^3)^2 = 4x^6 \]
Negative Exponents
A negative exponent means take the reciprocal. It does not mean the result is negative:
\[ a^{-n} = \frac{1}{a^n} \]
for \(a\neq0\).
For example:
\[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]
\[ 10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001 \]
Fractional Exponents
Fractional exponents connect powers and roots. For a positive base (a),
\[ a^{1/n} = \sqrt[n]{a} \]
and
\[ a^{m/n} = \sqrt[n]{a^m} \]
For example:
\[ 16^{1/2} = \sqrt{16} = 4 \]
\[ 16^{3/2} = (\sqrt{16})^3 = 4^3= 64 \]
Another example is:
\[ 27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9 \]
Negative bases need extra care in real-number math. An odd root of a negative number is real:
\[ (-8)^{1/3} = -2 \]
But an even root of a negative number is not a real number:
\[ (-16)^{1/2} \]
has no real value.
Do not confuse a fractional exponent with a fractional base:
\[ \left(\frac14\right)^3 = \frac1{64} \]
Here, \(\frac14\) is the base and \(3\) is the exponent.
Solving Exponential Equations
An expression such as \(5^4\) is evaluated. An equation such as \(2^x=32\) is solved for an unknown value.
Rewrite both sides with the same base when possible:
\[ 2^x = 32 \]
\[ 2^x = 2^5 \]
\[ x = 5 \]
Not every exponential equation can be solved by matching bases. More complicated equations may require logarithms, graphing, or another algebraic method. For equations with an unknown variable, try the Solve for x Calculator.
Scientific Notation and Exponents
Scientific notation uses powers of \(10\) to write large and small numbers.
\[ 4.2 \times 10^5 = 420000 \]
The exponent \(5\) moves the decimal point five places right.
A negative exponent moves it left:
\[ 3.5 \times 10^{-4} = 0.00035 \]
Types of Exponents
These categories can overlap. For example, \(-3\) is both a negative exponent and an integer exponent.
Positive Exponents
A positive integer exponent represents repeated multiplication:
\[ 4^3 = 4 \times 4 \times 4 = 64 \]
Exponent of 1
An exponent of \(1\) leaves the base unchanged:
\[ a^1 = a \]
for every defined value of \(a\).
Zero Exponents
Any nonzero base raised to the zero power equals \(1\):
\[ a^0 = 1 \]
for \(a\neq0\).
The expression \(0^0\) is not assigned a universal value by the basic exponent rule. Its treatment depends on the mathematical context.
Negative Exponents
A negative exponent gives the reciprocal of the corresponding positive power:
\[ a^{-n} = \frac1{a^n} \]
for \(a\neq0\).
Integer Exponents
Integer exponents include positive integers, zero, and negative integers. Examples include \(x^5\), \(x^0\), and \(x^{-2}\).
Fractional or Rational Exponents
A rational exponent has the form \(\frac{m}{n}\), where \(n\neq0\). For positive real bases, it represents a root and a power:
\[ a^{m/n} = \left(\sqrt[n]{a}\right)^m \]
For negative real bases, a real result is possible when the reduced denominator is odd. An even denominator requires an even root and does not give a real result for a negative base.
Variable Exponents
In \(2^x\), the variable \(x\) is the exponent. This differs from \(x^2\), where the variable is the base. Variable exponents appear in exponential functions and equations.
Laws and Rules of Exponents
Exponent laws apply when the structure of the expression matches the rule. Check the bases, parentheses, and denominators before using one.
Product of Powers
\[ a^m a^n = a^{m+n} \]
Add exponents only when multiplying powers with the same base.
\[ y^2y^7 = y^9 \]
Quotient of Powers
\[ \frac{a^m}{a^n} = a^{m-n} \]
for \(a\neq0\).
\[ \frac{z^9}{z^4} = z^5 \]
Power of a Power
\[ (a^m)^n = a^{mn} \]
Multiply the exponents:
(x^3)^2 = x^6
Power of a Product
\[ (ab)^n = a^n b^n \]
For example:
\[ (2x)^3 = 2^3x^3 = 8x^3 \]
Power of a Quotient
\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]
for \(b\neq0\).
\[ \left(\frac{x}{3}\right)^2 = \frac{x^2}{9} \]
Zero and Negative Exponents
\[ a^0 = 1 \]
for \(a\neq0\), and
\[ a^{-n} = \frac1{a^n} \]
for \(a\neq0\).
An exponent does not distribute across addition or subtraction:
\[ (a+b)^n\neq a^n+b^n \]
in general.
Also, exponents cannot be combined just because two powers are multiplied:
\[ 2^3\cdot3^4 \]
does not become \(2^7\) or \(3^7\), because the bases are different.
Simplifying Expressions with Exponents
Simplifying rewrites an expression in an equivalent, cleaner form. It is different from solving an equation for a variable. For broader algebraic simplification, use the Simplify Calculator.
Consider:
\[ \frac{2^5}{2^7} \]
Apply the quotient rule:
\[ 2^{5-7} = 2^{-2} \]
Then rewrite the negative exponent using a positive exponent:
\[ 2^{-2} = \frac1{2^2} = \frac14 \]
Now simplify an expression with coefficients and variables:
\[ \frac{18x^6}{3x^2} \]
\[ \frac{18}{3}\cdot\frac{x^6}{x^2} = 6x^{6-2} \]
\[ 6x^4 \]
This requires \(x\neq0\), because \(x^2\) appears in the original denominator.
A second example uses two rules:
\[ \frac{(2x^3)^2}{4x} \]
\[ \frac{4x^6}{4x} = x^{6-1} \]
\[ x^5 \]
The original denominator requires \(x\neq0\).
Exponential Functions and Equations
An exponential function has a variable in its exponent:
\[ y = 2^x \]
As \(x\) increases by \(1\), the output is multiplied by \(2\).
Compare that with a power function:
\[ y = x^2 \]
In \(y=2^x\), the variable is the exponent. In \(y=x^2\), the variable is the base.
An exponential equation asks for a value that makes two expressions equal:
\[ 2^x = 16 \]
Since \(16=2^4\),
\[ 2^x = 2^4 \]
so
\[ x = 4 \]
Concept Explainer
For a positive integer exponent, the exponent tells how many times the base appears as a factor. In \(5^4\), the base is \(5\), and the exponent is \(4\).
A power is the full expression made from a base and an exponent:
\[ 5^4 = 5 \times 5 \times 5 \times 5 = 625 \]
Exponents can also be negative or fractional. Negative exponents represent reciprocals, while fractional exponents represent roots when a real-valued result exists. Variables can be bases, as in \(x^2\), or exponents, as in \(2^x\).
The expression \(5^4\) has one numerical value, so you evaluate it. The equation \(2^x=16\) asks for the value of \(x\), so you solve it.
Worked Example
Worked Example 1 — Numerical Power
Find \(8^5\).
\[ 8^5= 8 \times 8 \times 8 \times 8 \times 8 \]
The exponent \(5\) means use \(8\) as a factor five times.
\[ 8 \times 8 = 64 \]
Multiply the first two factors.
\[ 64 \times 8 = 512 \]
Multiply by the third factor.
\[ 512 \times 8 = 4096 \]
Multiply by the fourth factor.
\[ 4096 \times 8 = 32768 \]
Multiply by the final factor.
\[ 8^5 = 32768 \]
So \(8\) to the fifth power equals \(32768\).
Worked Example 2 — Variable Expression
Simplify:
\[ \frac{12x^7}{3x^2} \]
\[ \frac{12x^7}{3x^2}=\frac{12}{3}\cdot\frac{x^7}{x^2} \]
Separate the coefficient and variable parts.
\[ \frac{12}{3} = 4 \]
Divide the coefficients.
\[ \frac{x^7}{x^2} = x^{7-2} \]
Use the quotient rule because both powers have base \(x\).
\[ x^{7-2} = x^5 \]
Subtract the exponents.
\[ \frac{12x^7}{3x^2} = 4x^5 \]
The simplified expression is \(4x^5\).
The restriction is \(x\neq0\), because \(x^2\) is in the original denominator.