Lesson 13-5: Explore Patterns of Exponents

Property

Any repeated multiplication can be written in exponential form: $a \cdot a \cdot a \cdot \ldots \cdot a$ ($n$ factors) = $a^n$.

Conversely, any exponential expression can be expanded back into repeated multiplication. The base ($a$) indicates what number is being multiplied, and the exponent ($n$) indicates how many times the base appears as a factor.

Examples

• $5 \cdot 5 \cdot 5 \cdot 5 = 5^4$ (four factors of 5)
• $x^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x$ (six factors of $x$)
• $(-3) \cdot (-3) \cdot (-3) = (-3)^3$ (three factors of -3)

Property

As the exponent of a base decreases by 1, the value of the power is divided by the base. Following this pattern past the exponent of 1 reveals the rules for zero and negative exponents:
For every nonzero number $x$, $x^0 = 1$.
For every nonzero number $x$, $x^{-n} = \frac{1}{x^n}$.

Examples

• Consider the pattern for powers of 2:

Notice that $2^{-2}$ is exactly the same as $\frac{1}{2^2}$.

• Simplify $\frac{x^{-4}}{y^2}$: Apply the "flip" rule to the negative exponent to move it to the denominator, resulting in $\frac{1}{x^4 y^2}$.

Explanation

Zero and negative exponents aren't magic; they are just the logical continuation of a mathematical pattern! Every time an exponent drops by one, you divide by the base. This proves why any non-zero number to the power of zero is exactly 1. It also shows that a negative exponent is basically a "flip-it" command: it tells you to take the reciprocal of the base and make the exponent positive.

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!