Lesson 1: Exponents

New Concept

Exponents provide a shorthand for repeated multiplication, where a base is raised to a power, like $a^n$. You'll learn how to compute powers, simplify expressions, and combine terms that include exponents.

What’s next

Now, you’ll tackle a series of practice cards and worked examples to master computing powers and applying the order of operations with exponents.

Property

An exponent is a number that appears above and to the right of a particular factor. It tells us how many times that factor occurs in the expression. The factor to which the exponent applies is called the base, and the product is called a power of the base.
An exponent indicates repeated multiplication.

where $n$ is a positive integer.

Examples

• To compute $5^3$, we multiply three factors of 5: $5 \cdot 5 \cdot 5 = 125$.

• The expression $(\frac{1}{4})^2$ means $\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}$.

Property

To show that a negative number is raised to a power, we enclose the negative number in parentheses. For example, to indicate the square of $-5$, we write

If the negative number is not enclosed in parentheses, the exponent applies only to the positive number.

Examples

• To calculate $(-3)^4$, we multiply four factors of $-3$: $(-3)(-3)(-3)(-3) = 81$.

• The expression $-3^4$ means the negative of $3^4$, so we calculate $3 \cdot 3 \cdot 3 \cdot 3 = 81$ and then apply the negative sign to get $-81$.

Property

An exponent on a variable indicates repeated multiplication, while a coefficient in front of a variable indicates repeated addition.

but

Examples

• For the variable $y$, the expression $y^3$ means $y \cdot y \cdot y$.

• For the same variable $y$, the expression $3y$ means $y+y+y$.

Property

1. Perform any operations inside parentheses, or above or below a fraction bar.
2. Compute all indicated powers.
3. Perform all multiplications and divisions in the order in which they occur from left to right.
4. Perform additions and subtractions in order from left to right.

Examples

• To simplify $10 - 3 \cdot 2^2$, we first compute the power: $10 - 3 \cdot 4$. Then multiply: $10 - 12 = -2$.

• In the expression $(4+1)^3 \div 5$, we first work inside the parentheses: $5^3 \div 5$. Then compute the power: $125 \div 5 = 25$.

Property

We can combine like powers of the same variable. When we add like terms, we do not alter the exponent; only the coefficient of the power changes. For example:

Different powers of the same variable are not like terms and cannot be combined. For example, $8x^2 - 3x$ cannot be simplified.

Examples

• The terms $7a^3$ and $4a^3$ are like terms, so they can be combined: $7a^3 - 4a^3 = 3a^3$.

• The expression $5w^2 + 3w^3$ cannot be simplified because $w^2$ and $w^3$ are not like terms.