Lesson 6: Exponents

Property

$a^n$ means multiply $n$ factors of $a$.

($n$ factors)
This is read $a$ to the $n^{\text{th}}$ power. In the expression $a^n$, the exponent tells us how many times we use the base $a$ as a factor.

Examples

• $4^3$ means $4 \cdot 4 \cdot 4$, which simplifies to $64$.
• $(-2)^4$ means $(-2)(-2)(-2)(-2)$, which is $16$. Be careful, as $-2^4$ means $-(2 \cdot 2 \cdot 2 \cdot 2)$, which is $-16$.
• $(\frac{1}{5})^2$ means $\frac{1}{5} \cdot \frac{1}{5}$, resulting in $\frac{1}{25}$.

Explanation

The exponent is a counter for how many times the base number is multiplied by itself.
The base is the number being multiplied, and the exponent is the small number written up high, telling you the count.

Property

The placement of parentheses completely changes the meaning and the result of an expression with a negative base.
In $(-a)^n$, the base is $-a$ and the entire negative number is multiplied $n$ times.
In $-a^n$, the base is just $a$; the exponent is applied first, and then the negative sign is attached to the final result.

Examples

• To simplify $(-3)^2$, the base is -3. You calculate $(-3)(-3) = 9$.
• To simplify $-3^2$, the base is 3. You calculate $3^2 = 9$ first, and then take the opposite, giving -9.
• Evaluate $k^2 - k$ for $k = -6$:

Substitute with parentheses: $(-6)^2 - (-6) = 36 + 6 = 42$.

Explanation

Parentheses act like a protective force field! When you write $(-5)^2$, you're telling the math world to square the entire thing inside, negative sign and all, resulting in a positive 25. But without that force field, $-5^2$ means you only square the 5, and the negative sign just waits outside to get tacked on at the very end. Always use parentheses when substituting negative numbers!

Property

$a^0 = 1$, if $a \neq 0$

This definition is based on the second law of exponents. For any non-zero number $a$, the quotient $\frac{a^n}{a^n}$ is equal to 1. Using the law of exponents, we can also write $\frac{a^n}{a^n} = a^{n-n} = a^0$. Therefore, it is logical to define $a^0$ as 1.

Examples

• For a positive integer, $8^0 = 1$.
• For a negative integer, $(-55)^0 = 1$.
• For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$.