Simplifying Exponents
This is a companion to a video where we simplify expressions that contain exponents and require different combinations of exponent rules to be simplified.
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 | $(a^m)^n=a^{mn}$ |
| Negative exponent | A factor has a negative power | $a^{-n}=\dfrac{1}{a^n}$ |
Simplification Example that Only Uses the Product Rule
Consider
\[x^5y^3x^2y^6\]Match up the factors with base $x$ together and use the product rule.
\[x^5x^2=x^{5+2}=x^7\]Similarly, match up the factors with base $y$ together and use the product rule.
\[y^3y^6=y^{3+6}=y^9\]So we get
\[x^5y^3x^2y^6 = x^7y^9\]Watch this part: product rule example at 0:17.
Simplification Example that Only Uses the Power Rule
Consider
\[\left(x^3(y+1)^2\right)^4\]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
\[\left((y+1)^2\right)^4=(y+1)^8\]So we get
\[\left(x^3(y+1)^2\right)^4 = x^{12}(y+1)^8\]Watch this part: power rule example at 4:03.
Simplification Example with Two Rules (Negative Exponent and Quotient Rule)
Consider
\[\frac{(x^3z)^2y^{-2}z^4}{y(x+1)}\]Start with the grouped power and apply the power rule
\[(x^3z)^2=x^6z^2\]Substitute that back into the expression
\[\frac{x^6z^2y^{-2}z^4}{y(x+1)}\]Now combine the $z$ factors in the numerator using the product rule
\[z^2z^4=z^6\]$y$ factors exist in the numerator and denominator so apply the quotient rule
\[\frac{y^{-2}}{y^1}=y^{-3}\]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:
\[\frac{(x^3z)^2y^{-2}z^4}{y(x+1)} = \frac{x^6z^6}{y^3(x+1)}\]Watch this part: mixed exponent example at 5:47.
Simplification Example with Three Rules (Quotient/Product/Power Rule)
Consider
\[\frac{(x^2yz)^3}{xy^4z^3y^8}\]Apply the power rule to the numerator
\[(x^2yz)^3=x^6y^3z^3\]The denominator has two $y$ factors, apply the product rule
\[y^4y^8=y^{12}\]So the expression becomes
\[\frac{x^6y^3z^3}{xy^{12}z^3}\]Now match bases in the numerator and denominator and apply the quotient rule.
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:
\[\frac{(x^2yz)^3}{xy^4z^3y^8} = \frac{x^5}{y^9}\]Watch this part: quotient and power rules at 8:18.
Simplification Example with ALL Rules
Consider
\[\frac{x^5(y^{-3}z^3)^{-4}(z+1)^5}{(z+1)^{-2}x^7z^9}\]Apply the power rule
\[(y^{-3}z^3)^{-4}=y^{12}z^{-12}\]Substituting that back in we get
\[\frac{x^5y^{12}z^{-12}(z+1)^5}{(z+1)^{-2}x^7z^9}\]Apply product rule when appropriate
For $x$ factors in numerator
\[\frac{x^5}{x^7}=x^{-2}=\frac{1}{x^2}\]For $y$ factor in numerator there is nothing to combine it wit.
\[y^{12}\]For factors of $z$ we use the quotient rule
\[\frac{z^{-12}}{z^9}=z^{-21}=\frac{1}{z^{21}}\]For factors of $z+1$ we use the quotient rule
\[\frac{(z+1)^5}{(z+1)^{-2}}=(z+1)^{5-(-2)}=(z+1)^7\]Putting the positive-power pieces together gives:
\[\frac{x^5(y^{-3}z^3)^{-4}(z+1)^5}{(z+1)^{-2}x^7z^9} = \frac{y^{12}(z+1)^7}{x^2z^{21}}\]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: