Calculator with exponent

April 22, 2025

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Exponent Calculator (Power Calculator)

Calculate the result of exponentiation, where a base number is raised to the power of an exponent. Enter the base (a) and the exponent (n) to compute $a^n$.

Calculate $a^n$

Base (a):

Exponent (n):

Understanding Exponents

Exponentiation is a fundamental mathematical operation, written as $\bf{a^n}$, involving two numbers: the base ($a$) and the exponent or power ($n$). When $n$ is a positive integer, exponentiation represents repeated multiplication of the base, $n$ times.

$a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ times}}$

Exponents can also be zero, negative, or fractional (represented as decimals in this calculator), each with specific rules.

Basic Exponent Laws and Rules

Product Rule:
$a^n \times a^m = a^{(n+m)}$
Multiply powers with the same base: add the exponents.
Ex: $2^2 \times 2^3 = 2^{(2+3)} = 2^5 = 32$

Quotient Rule:
$\frac{a^m}{a^n} = a^{(m - n)}$
Divide powers with the same base: subtract the exponents.
Ex: $3^5 / 3^2 = 3^{(5-2)} = 3^3 = 27$

Power Rule:
$(a^m)^n = a^{(m \times n)}$
Raise a power to another power: multiply the exponents.
Ex: $(4^2)^3 = 4^{(2 \times 3)} = 4^6 = 4096$

Power of a Product Rule:
$(a \times b)^n = a^n \times b^n$
Raise a product to a power: distribute the exponent.
Ex: $(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$

Power of a Quotient Rule:
$(\frac{a}{b})^n = \frac{a^n}{b^n}$ (where $b \neq 0$)
Raise a quotient to a power: distribute the exponent.
Ex: $(2/5)^2 = 2^2 / 5^2 = 4/25$

Negative Exponent Rule:
$a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$)
Negative exponent means reciprocal of the base to the positive exponent.
Ex: $2^{-3} = 1 / 2^3 = 1/8$

Zero Exponent Rule:
$a^0 = 1$ (where $a \neq 0$)
Any non-zero base raised to the power of 0 equals 1.
Ex: $5^0 = 1$, $(-3)^0 = 1$

Exponent of One Rule:
$a^1 = a$
Any base raised to the power of 1 is the base itself.
Ex: $7^1 = 7$

Fractional Exponent Rule (Roots):
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
Denominator is root, numerator is power. Input decimals (e.g., 0.5 for square root).
Ex: $8^{1/3} = \sqrt[3]{8} = 2$. $9^{3/2} = (\sqrt{9})^3 = 3^3 = 27$.

Calculator Capabilities and Limitations

Accepts positive and negative bases.
Accepts positive, negative, and zero exponents.
Accepts decimal exponents (for fractional powers). Enter fractions like 1/2 as 0.5, 2/3 as 0.6666... etc.
Does NOT compute imaginary numbers. If you enter a negative base with a non-integer exponent (e.g., $(-4)^{0.5}$), the result will be indicated as invalid or "NaN" (Not a Number), as this involves imaginary units ($i$).

Examples with Step-by-Step Solutions

Click on each example to see the calculation:

Example 1: Positive Integer Exponent ($5^3$)

Problem: Calculate $5^3$.

Steps:

Identify the base ($a=5$) and the exponent ($n=3$).
Since the exponent is a positive integer, multiply the base by itself 3 times.
Calculation: $5 \times 5 \times 5 = 25 \times 5 = 125$.

Result: $5^3 = 125$.

Example 2: Negative Integer Exponent ($4^{-2}$)

Problem: Calculate $4^{-2}$.

Steps:

Apply the Negative Exponent Rule: $a^{-n} = 1/a^n$.
Here, $a=4$ and $n=2$. So, $4^{-2} = 1 / 4^2$.
Calculate the denominator: $4^2 = 4 \times 4 = 16$.
Calculation: $1 / 16$.

Result: $4^{-2} = 1/16 = 0.0625$.

Example 3: Zero Exponent ($(-7)^0$)

Problem: Calculate $(-7)^0$.

Steps:

Apply the Zero Exponent Rule: $a^0 = 1$ (for any non-zero base $a$).
The base is $a=-7$, which is non-zero.

Result: $(-7)^0 = 1$.

Example 4: Exponent of One ($10^1$)

Problem: Calculate $10^1$.

Steps:

Apply the Exponent of One Rule: $a^1 = a$.
The base is $a=10$.

Result: $10^1 = 10$.

Example 5: Decimal/Fractional Exponent ($9^{1.5}$)

Problem: Calculate $9^{1.5}$. (Note: 1.5 = 3/2)

Steps using Fractional Exponent Rule ($a^{m/n} = (\sqrt[n]{a})^m$):

Identify $a=9$, $m=3$, $n=2$.
Calculate the nth root: $\sqrt[2]{9} = \sqrt{9} = 3$.
Raise the result to the mth power: $3^3 = 3 \times 3 \times 3 = 27$.

Steps using Decimal (Calculator Method):

Enter Base = 9, Exponent = 1.5 into the calculator.
The calculator computes $9^{1.5}$.

Result: $9^{1.5} = 27$.

Example 6: Negative Base, Even Exponent ($(-3)^4$)

Problem: Calculate $(-3)^4$.

Steps:

Multiply the base $(-3)$ by itself 4 times.
Calculation: $(-3) \times (-3) \times (-3) \times (-3)$
$(9) \times (-3) \times (-3)$
$(-27) \times (-3)$
$81$ (Negative times negative is positive).

Result: $(-3)^4 = 81$. (Note: The result is positive because the exponent is even).

Example 7: Negative Base, Odd Exponent ($(-2)^3$)

Problem: Calculate $(-2)^3$.

Steps:

Multiply the base $(-2)$ by itself 3 times.
Calculation: $(-2) \times (-2) \times (-2)$
$(4) \times (-2)$
$-8$ (Positive times negative is negative).

Result: $(-2)^3 = -8$. (Note: The result is negative because the exponent is odd).

Example 8: Product Rule ($2^2 \times 2^3$)

Problem: Calculate $2^2 \times 2^3$.

Steps using Product Rule:

The bases are the same (2). Add the exponents: $2 + 3 = 5$.
Result is $2^5$.
Calculate $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

Steps direct calculation:

$2^2 = 4$.
$2^3 = 8$.
$4 \times 8 = 32$.

Result: $2^2 \times 2^3 = 32$.

Example 9: Quotient Rule ($10^6 / 10^4$)

Problem: Calculate $10^6 / 10^4$.

Steps using Quotient Rule:

The bases are the same (10). Subtract the exponents: $6 - 4 = 2$.
Result is $10^2$.
Calculate $10^2 = 10 \times 10 = 100$.

Steps direct calculation:

$10^6 = 1,000,000$.
$10^4 = 10,000$.
$1,000,000 / 10,000 = 100$.

Result: $10^6 / 10^4 = 100$.

Example 10: Power Rule ($(4^2)^3$)

Problem: Calculate $(4^2)^3$.

Steps using Power Rule:

Multiply the exponents: $2 \times 3 = 6$.
Result is $4^6$.
Calculate $4^6 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096$.

Steps direct calculation:

Calculate inside the parenthesis: $4^2 = 16$.
Calculate $16^3 = 16 \times 16 \times 16 = 256 \times 16 = 4096$.

Result: $(4^2)^3 = 4096$.

Frequently Asked Questions (FAQs)

What is the 'base' and 'exponent'?

In the expression $a^n$, '$a$' is the base (the number being multiplied) and '$n$' is the exponent (how many times the base is multiplied by itself, or other interpretations based on the rules).

How does this calculator handle $a^{-n}$?

It applies the rule $a^{-n} = 1/a^n$. For example, inputting Base=2, Exponent=-3 will calculate $1 / 2^3 = 1/8 = 0.0625$.

How do I enter fractional exponents like $8^{2/3}$?

You need to enter the exponent in its decimal form. For $2/3$, you would enter approximately 0.6666667 in the exponent field. For $1/2$, enter 0.5.

What happens with negative bases like $(-4)^{0.5}$?

Calculating a non-integer power (like 0.5, which is the square root) of a negative number results in an imaginary number (involving $i = \sqrt{-1}$). This calculator does not compute imaginary numbers and will display an error or "Invalid Input" message in such cases.

What result does the calculator give for $0^0$?

$0^0$ is mathematically indeterminate, meaning it doesn't have a single defined value. However, in many programming contexts and for certain mathematical conveniences (like in binomial expansions), it is often defined as 1. This calculator uses the standard JavaScript `Math.pow(0, 0)` function, which returns 1.

Where are exponents used?

Exponents are fundamental in many fields, including: science (scientific notation, exponential growth/decay), finance (compound interest calculations), computer science (data sizes, algorithms), statistics, engineering, and many other areas involving repeated multiplication or scaling relationships.