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Exponent Laws Review

The Rules of Exponents

Exponents are a compact way to write out repeated multiplication. Instead of writing $x cdot x cdot x cdot x cdot x$, we can just write $x^5$. The laws of exponents are shortcuts that help us simplify expressions without having to write out all the terms. These rules are the foundation for working efficiently with large and small numbers, especially in scientific notation.

Combining Like Bases

The first two rules apply when you multiply or divide expressions that share the same base.

When multiplying terms with the same base, you add their exponents.

amcdotan=am+na^m \\ cdot a^n = a^{m+n}

Think about what this means. If you have $x^2 cdot x^3$, that's just $(x cdot x) cdot (x cdot x cdot x)$, which is five $x$'s multiplied together, or $x^5$. Adding the exponents gets you there faster: $x^{2+3} = x^5$.

The opposite is true for division.

When dividing terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

aman=am−n\frac{a^m}{a^n} = a^{m-n}

For example, $ frac{y^7}{y^4}$ means we have seven $y$'s on top and four on the bottom. Four pairs will cancel out, leaving three $y$'s on top. This is the same as $y^{7-4} = y^3$.

Powers and Reciprocals

What happens when you raise an exponential term to another power? Or when the exponent is negative or zero? These situations have their own straightforward rules.

To raise a power to another power, you multiply the exponents.

(am)n=amcdotn(a^m)^n = a^{m \\ cdot n}

Let's look at (x4)3(x^4)^3. This means we are multiplying x4x^4 by itself three times: x4cdotx4cdotx4x^4 cdot x^4 cdot x^4. Using the product rule from before, we add the exponents: x4+4+4=x12x^{4+4+4} = x^{12}. The shortcut is to multiply them directly: x4cdot3=x12x^{4 cdot 3} = x^{12}.

Now, let's consider negative exponents. A negative exponent doesn't make the number negative. Instead, it signifies a reciprocal.

a−n=frac1ana^{-n} = \\ frac{1}{a^n}

If you have $2^{-3}$, it means $ frac{1}{2^3}$, which equals $ frac{1}{8}$. Moving an exponential term from the numerator to the denominator (or vice versa) flips the sign of its exponent. So, $ frac{1}{x^{-5}}$ is the same as $x^5$.

Finally, there's the zero exponent. This one is simple.

Any non-zero base raised to the power of zero is equal to 1.

a0=1quad(foraneq0)a^0 = 1 \\ quad (for \\ a \\ neq 0)

Why? We can use the quotient rule to see this. What is $ frac{x^5}{x^5}$? Any number divided by itself is 1. Using the quotient rule, we get $x^{5-5} = x^0$. Therefore, $x^0$ must equal 1.

Let's test your understanding of these rules.

Quiz Questions 1/5

Simplify the expression x3⋅x7x^3 \cdot x^7.

Quiz Questions 2/5

What is the value of any non-zero number raised to the power of zero?

Mastering these exponent laws is essential for simplifying complex algebraic expressions and is a key skill for working with scientific notation.