Exponents
When simplifying exponents, remember the exponentiation is just repeated multiplications. So if you have something like
[latex]x^3 x^7[/latex]
This is three [latex]x[/latex]s multiplied by seven [latex]x[/latex]s, so that’s ten [latex]x[/latex]s all multiplied together.
[latex]x^3 x^7 = x^{10}[/latex]
Similarly, all these other rules don’t even have to be memorized if you just think about how repeated multiplication would work. But here they are anyway.
\begin{align*}
A^x A^y & = A^{x + y} \\
\frac{A^x}{A^y} & = A^{x – y} \\
\left( A^x \right)^y & = A^{xy} \\
A^{-x} & = \frac{1}{A^x} \\
A^0 & = 1
\end{align*}
Some examples:
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[latex]\frac{e^{11}}{e^5 e^4}[/latex].
Here we have eleven [latex]e[/latex]s, and we are taking away via division five of them then four of them. Hence we have two [latex]e[/latex]s left over: [latex]\frac{e^{11}}{e^5 e^4} = \boxed{e^2}[/latex]. Note that [latex]e[/latex] is a fundamental constant in mathematics, like [latex]\pi[/latex], equal to [latex]2.718281828459045\ldots[/latex] approximately, but we just use [latex]e[/latex] for the exact value.
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[latex]\frac{(A^4 B)^3}{(A B^4)^2}[/latex].
We see [latex](A^4 B)^3 = A^{12} B^3[/latex]. On bottom, we have [latex](A B^4)^2 = A^2 B^8[/latex]. This give [latex]\frac{A^{12} B^3}{A^2 B^8}[/latex]. Once we get cancel two of the As and three of the Bs, we have [latex]\boxed{\frac{A^{10}}{B^5}}[/latex].
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[latex]\left(\frac{\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}}{\sqrt[4]{104942}} \right)^{x - x}[/latex].
This looks ugly, but [latex]x - x = 0[/latex], and anything to the zeroth power is [latex]1[/latex]. Hence the answer is [latex]\boxed{1}[/latex].
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[latex]a^{-5}a^2[/latex].
This combines as [latex]a^{-5 + 2} = a^{-3}[/latex], which we can also write as [latex]\boxed{\frac{1}{a^3}}[/latex].
Fractional Exponents
One more rule before you go: [latex]A^{n/m} = \left(\sqrt[m]{A}\right)^n[/latex]. In other words, a fraction in the exponent is the same thing as taking a square root, cube root, 4th root, etc, depending on what the denominator is. Some examples:
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[latex]25^{1/2}[/latex].
We see this is the same thing as [latex]\sqrt{25}[/latex], which is [latex]\boxed{5}[/latex].
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[latex]8^{2/3}[/latex].
This is the same thing as [latex]\left( \sqrt[3]{8}\right)^2[/latex]. We see that [latex]\sqrt[3]{8} = 2[/latex], and hence we do [latex]2^2 = 4[/latex]. So [latex]8^{2/3} = \boxed{4}[/latex].