Lesson 3: Using Rules of Exponents

New Concept

If $n$ is any real number and $x$ is any real number that is not zero, $x^{-n} = \frac{1}{x^n}$.

What’s next

Next, you’ll use this rule, along with product and power rules, to simplify complex algebraic expressions and numbers in scientific notation.

If n is any real number and x is any real number that is not zero,

To simplify $5^{-2}$, move the base to the denominator and make the exponent positive: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$. The negative sign matters! $-4^{-2} = -\frac{1}{16}$, but with parentheses, $(-4)^{-2} = \frac{1}{(-4)^2} = \frac{1}{16}$. If the negative exponent is in the denominator, move it to the numerator: $\frac{1}{3^{-2}} = 3^2 = 9$.

A negative exponent is a secret signal! It doesn't make the number negative. It just tells the base to move to the other side of the fraction bar, where its exponent then becomes positive. Flip it and switch the sign!

If m, n, and x are real numbers and $x \neq 0$,

Combine powers of the same base by adding their exponents: $a^4 \cdot a^3 = a^{4+3} = a^7$. This works with negative exponents too: $b^5 \cdot b^{-2} = b^{5+(-2)} = b^3$. In mixed expressions, group like bases first: $x^3 y^2 x^{-1} y^4 = (x^3 x^{-1})(y^2 y^4) = x^2 y^6$.

When multiplying terms with the same base, you're just combining groups of factors. Skip the long-hand work and take a shortcut! Just add the exponents together to find the new total number of factors.

Property

If m, n, and x are real numbers, the Power of a Power property states:

This rule extends to products inside parentheses (Power of a Product property), meaning the outside power applies to every factor inside:

Examples

• Power of a Power: To simplify $(x^4)^3$, you multiply the exponents: $(x^4)^3 = x^{4 \cdot 3} = x^{12}$.
• Power of a Product: Apply the outside exponent to every factor inside the parentheses: $(a^2 b^5)^3 = (a^2)^3 (b^5)^3 = a^6 b^{15}$.
• With Negative Exponents: This rule works perfectly with negative exponents as well: $(y^{-3})^2 = y^{-3 \cdot 2} = y^{-6} = \frac{1}{y^6}$.

Explanation

Raising a power to another power is like making copies of copies! You have 'n' groups, and each group contains 'm' factors. To get the total number of factors, you simply multiply the two exponents together. It's the ultimate power-up move for your math skills, allowing you to bypass writing out huge strings of variables.