Lesson 2: Higher Exponents

Property

For any expression $a^n$, $a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.
The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$, the $a$ is called the base and the $n$ is called the exponent.
$a^n = a \cdot a \cdot a \cdot ... \cdot a$ ($n$ factors)
Special names are used for powers of 2 and 3:
$a^2$ is read as '$a$ squared'
$a^3$ is read as '$a$ cubed'

Examples

• The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$, where 5 is the base and 4 is the exponent.
• To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$.
• To simplify $2^5$, you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, which equals $32$.

Explanation

Exponents are a shortcut for writing repeated multiplication. The small exponent number tells you how many times to multiply the larger base number by itself. This makes it much faster to write out long calculations involving the same factor.

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

To raise a product to a power, raise each factor to the power. In symbols,

Examples

• To simplify $(4xy)^2$, apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$.
• For $(-3a^2)^3$, raise each factor to the third power: $(-3)^3(a^2)^3 = -27a^6$.
• Note the difference: in $5x^3$, only $x$ is cubed. In $(5x)^3$, both 5 and $x$ are cubed, giving $125x^3$.

Explanation

This rule works because multiplication is commutative. An expression like $(2x)^3$ means $(2x)(2x)(2x)$. You can regroup the factors as $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x)$, which is simply $2^3x^3$.