Multiplying exponents is simple with the \textbf{product of powers rule}: if two terms have the same base, add their exponents.

Product of Powers Rule

If two expressions share the same base, we add the exponents:

\[\begin{align*} a^m \cdot a^n = a^{m+n} \end{align*}\]

Example:

\[\begin{align*} x^2 \cdot x^5 = x^{2+5} = x^7 \end{align*}\]

Why It Works

Exponents represent repeated multiplication. Expanding \( x^2 \cdot x^3 \), we get:

\[\begin{align*} x^2 \cdot x^3 = (x \cdot x) \cdot (x \cdot x \cdot x) = x^5. \end{align*}\]

The first term contributes $2$ $x$’s, and the second term contributes $3$ $x$’s. The total number of $x$’s is $2 + 3 = 5$, which becomes the exponent.

Multiplying Multiple Exponents

The rule applies to more than two terms.

\[\begin{align*} x^2 \cdot x^4 \cdot x^3 &= (x \cdot x) \cdot (x \cdot x \cdot x \cdot x) \cdot (x \cdot x \cdot x) \\ &= x^{2+4+3} = x^9 \end{align*}\]

Summary

When multiplying exponents with the same base, we simply add the exponents:

\[\begin{align*} a^m \cdot a^n = a^{m+n} \end{align*}\]

This can be generalized to exponents that are not whole numbers. For example,
$$a^{1/2} \cdot a^{1/2} = a^{1/2 + 1/2} = a^1 = a.$$
This tells us that $a^{1/2}$ is a number that, when multiplied by itself, becomes $a$. We can see that $a^{1/2}$ is equivalent to the square root of $a$, or $\sqrt{a}$.

The product of powers rule introduces the concept of fractional exponents and even negative exponents. For more information or practice with exponents and related math topics, check out the “I Aced Calculus” app!