What an Exponent Counts
An exponent is a compact way to record repeated multiplication. Instead of writing the same factor many times, we write the factor once as the base and use an exponent to show how many copies of it are multiplied together.
For example, \(5^3\) means three factors of \(5\), multiplied together. It does not mean \(5\cdot 3\). The exponent counts the copies of the base; it is not itself a factor in the product.
The exponent \(1\) is included in the definition: \(a^1\) is a product with one factor, so \(a^1=a\). This definition does not yet specify powers with exponent zero or a negative exponent; those require separate definitions.
Reading and Evaluating a Power
To evaluate \(a^n\), identify the entire base, write it down exactly \(n\) times as a product, and then multiply. Writing the product before calculating is a useful way to check that the exponent has been interpreted correctly.
Include any parentheses that group the base.
The positive integer exponent tells you how many copies of that base to write.
Multiply the factors in order and confirm that the written product contains exactly the required number of copies.
Worked Example: Evaluating a Positive Base
Evaluate \(4^3\).
The base is \(4\), and the exponent is \(3\), so the product has exactly three factors of \(4\):
Multiply the first two factors, then multiply by the third:
Therefore, \(4^3=64\). To check, expand the answer back into the defining product: \(4\cdot4\cdot4=16\cdot4=64\), which has three factors of \(4\), as required.
The same definition applies when the base is a fraction or zero. For a fraction, multiply copies of the whole fraction. For zero, any positive number of zero factors has product zero.
Worked Example: A Fractional Base
Evaluate \(\left(\frac{2}{3}\right)^3\).
The parentheses show that the entire fraction is the base. The exponent \(3\) means to multiply three copies of that fraction:
Multiply the numerators and denominators:
Thus \(\left(\frac{2}{3}\right)^3=\frac{8}{27}\). Checking by expanding the result gives \(\frac{8}{27}=\frac{2\cdot2\cdot2}{3\cdot3\cdot3}\), the same three-factor product.
Negative Bases and the Number of Factors
A negative number can also be the base of a positive integer power. Parentheses make clear that the negative number, including its sign, is the factor being repeated. For example, \((-3)^4\) means \((-3)\cdot(-3)\cdot(-3)\cdot(-3)\).
The sign of the product depends on whether there are an even or an odd number of negative factors. Two negative factors multiply to a positive number. Grouping negative factors into pairs therefore explains the sign pattern for any positive integer exponent.
Here is a justification. The product defining \((-b)^n\) has \(n\) negative factors. If \(n\) is even, the factors can be grouped into pairs \((-b)(-b)\), and every pair has positive product. The product of those positive pairs is positive. If \(n\) is odd, pair all but one of the factors; the pairs have positive product, and the one unpaired factor is negative. The full product is therefore negative.
Worked Example: An Even Power of a Negative Base
Evaluate \((-3)^4\).
The parentheses make \(-3\) the base. There are four factors:
Group the factors into two pairs. Each pair has product \(9\):
so the full product is \(9\cdot9=81\). Thus \((-3)^4=81\). The positive result agrees with the sign rule because the exponent \(4\) is even. Checking the expansion gives four factors of \(-3\), and their product \(9\cdot9\) is \(81\).
Worked Example: An Odd Power of a Negative Base
Evaluate \((-2)^5\).
The base is \(-2\), and the exponent \(5\) requires five copies:
Pair four factors and leave one factor unpaired:
The product of the five factors is then \(4\cdot4\cdot(-2)=16\cdot(-2)=-32\). Therefore \((-2)^5=-32\). There are five factors, and the odd number of negative factors gives a negative result, as expected.
Parentheses Change the Base
Parentheses are especially important when a minus sign appears next to a power. In \((-3)^4\), the base is the entire number \(-3\), so the negative sign is repeated four times. In \(-3^4\), the exponent applies to \(3\), and the minus sign is outside the power. Under the standard order of operations, the exponent is evaluated before that outside negative sign.
These expressions are not interchangeable. The first has four factors of \(-3\); the second is the negative of a product with four factors of \(3\). Parentheses specify which number is the base, so read them before multiplying.
Worked Example: Distinguishing the Base from an Outside Sign
Evaluate \((-2)^3\) and \(-2^3\).
For the first expression, the base is \(-2\). Write three copies and multiply:
For the second expression, the base is \(2\); the minus sign is outside the power. Evaluate the power first:
In this particular case both expressions equal \(-8\), but their structures are different. The check is to expand each one: the first contains three factors of \(-2\), while the second has an outside negative sign and three factors of \(2\). The matching numerical answers here do not make the expressions equivalent in general.
A Common Mistake
A frequent error is to treat the exponent as a multiplier. For example, the incorrect calculation \(4^3=4\cdot3=12\) uses the exponent as if it meant “multiply the base by three.” But the definition says that the exponent counts copies of the base. The correct expansion is \(4^3=4\cdot4\cdot4=64\).
Another common error is to assume a negative sign always makes a power negative. That is only true for a negative base raised to an odd positive integer. With an even number of negative factors, the factors pair to give a positive product. Also check whether parentheses include the negative sign: \((-a)^n\) and \(-a^n\) have different structures.
Why This Notation Matters
Repeated multiplication appears throughout algebra and calculus. Powers let us write products compactly, compare quantities, and describe expressions without listing every factor. The notation is reliable only when the base and the number of repeated factors are interpreted precisely. In particular, knowing what a power means makes it possible to distinguish a power from multiplication and to reason correctly about signs.
This tutorial has defined powers with positive integer exponents. The exponent-zero and negative-exponent cases need their own definitions; they are not obtained by writing zero or a negative number of factors. Keeping that boundary clear prevents the repeated-multiplication definition from being applied where it does not apply.
Check Your Understanding
For each question, begin by identifying the base and writing the repeated product where appropriate.
- Write \(6^3\) as a product of factors, then evaluate it.
- Evaluate \((-4)^3\). How many negative factors are present, and what determines the sign of the result?
- Evaluate \((-5)^2\) and explain why its sign is positive.
- Explain the difference between \((-2)^4\) and \(-2^4\), then evaluate both.
- What does \(\left(\frac{1}{2}\right)^3\) mean as a product, and what is its value?
