Simplifying Exponents
This page follows the JoeCMath video on simplifying exponent expressions. The video starts with a direct product-rule example and builds toward expressions where powers, quotients, negative exponents, and grouped bases all have to be handled in the same problem.
Main Idea
When trying to simplify exponent expressions, it makes sense to focus on one base at a time (when applicable) or focus on leveraging one exponent rule at a time.
| Rule | When to use it | Pattern |
|---|---|---|
| Product rule | Same base multiplied together | $a^m a^n=a^{m+n}$ |
| Quotient rule | Same base divided | $\dfrac{a^m}{a^n}=a^{m-n}$ |
| Power rule | A power raised to another power | $(a^m)^n=a^{mn}$ |
| Power of a product | A whole product raised to a power | $(ab)^n=a^n b^n$ |
| Negative exponent | A factor has a negative power | $a^{-n}=\dfrac{1}{a^n}$ |
Product Rule
The first timestamped example from this video is:
\[x^5y^3x^2y^6\]There are two bases that repeat: $x$ and $y$. Combine only matching bases.
\[x^5x^2=x^{5+2}=x^7\]and
\[y^3y^6=y^{3+6}=y^9\]So the expression simplifies to:
\[x^7y^9\]Watch this part: product rule example at 0:17.
Powers On Parentheses
A later example asks you to distribute the outside exponent:
\[\left(x^3(y+1)^2\right)^4\]The outside power applies to each factor inside the parentheses:
\[\left(x^3\right)^4\left((y+1)^2\right)^4\]Now use the power rule on each factor.
\[\left(x^3\right)^4=x^{12}\]and
\[\left((y+1)^2\right)^4=(y+1)^8\]The simplified form is:
\[x^{12}(y+1)^8\]Watch this part: power rule example at 4:03.
Negative Exponents In A Quotient
One of the mixed examples is:
\[\frac{(x^3z)^2y^{-2}z^4}{y(x+1)}\]Start with the grouped power:
\[(x^3z)^2=x^6z^2\]Substitute that back into the expression:
\[\frac{x^6z^2y^{-2}z^4}{y(x+1)}\]Now combine the $z$ factors in the numerator:
\[z^2z^4=z^6\]The $y$ factors are split between the top and bottom. Since the denominator has $y=y^1$, subtract the exponents:
\[\frac{y^{-2}}{y^1}=y^{-3}\]A negative exponent means the factor belongs in the denominator with a positive exponent:
\[y^{-3}=\frac{1}{y^3}\]So the simplified expression is:
\[\frac{x^6z^6}{y^3(x+1)}\]Watch this part: mixed exponent example at 5:47.
Cancel After Expanding Powers
Another example from the video is:
\[\frac{(x^2yz)^3}{xy^4z^3y^8}\]First expand the numerator with the power of a product rule:
\[(x^2yz)^3=x^6y^3z^3\]The denominator has two $y$ factors:
\[y^4y^8=y^{12}\]So the expression becomes:
\[\frac{x^6y^3z^3}{xy^{12}z^3}\]Now subtract exponents for each matching base.
For $x$:
\[\frac{x^6}{x}=x^{6-1}=x^5\]For $y$:
\[\frac{y^3}{y^{12}}=y^{3-12}=y^{-9}=\frac{1}{y^9}\]For $z$:
\[\frac{z^3}{z^3}=z^{3-3}=z^0=1\]The simplified form is:
\[\frac{x^5}{y^9}\]Watch this part: quotient and power rules at 8:18.
Full Rule Mix
The hardest timestamped expression from the site archives is:
\[\frac{x^5(y^{-3}z^3)^{-4}(z+1)^5}{(z+1)^{-2}x^7z^9}\]Start with the power on the grouped product:
\[(y^{-3}z^3)^{-4}=y^{12}z^{-12}\]Now rewrite the expression with that part simplified:
\[\frac{x^5y^{12}z^{-12}(z+1)^5}{(z+1)^{-2}x^7z^9}\]Combine by base.
For $x$:
\[\frac{x^5}{x^7}=x^{-2}=\frac{1}{x^2}\]For $y$:
\[y^{12}\]For $z$:
\[\frac{z^{-12}}{z^9}=z^{-21}=\frac{1}{z^{21}}\]For the grouped base $z+1$:
\[\frac{(z+1)^5}{(z+1)^{-2}}=(z+1)^{5-(-2)}=(z+1)^7\]Putting the positive-power pieces together gives:
\[\frac{y^{12}(z+1)^7}{x^2z^{21}}\]Watch this part: full rule mix at 11:01.
Timestamp Guide
More Practice
For the full set of timestamped exponent examples already organized on the site, use the existing archives: