3.1: Algebra Tips and Tricks Part V (Exponents)
Exponents
When simplifying exponents, remember the exponentiation is just repeated multiplications. So if you have something like
\(x^3 x^7\)
This is three \(x\)s multiplied by seven \(x\)s, so that’s ten \(x\)s all multiplied together.
\(x^3 x^7 = x^{10}\)
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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\(\frac{e^{11}}{e^5 e^4}\).
Here we have eleven \(e\)s, and we are taking away via division five of them then four of them. Hence we have two \(e\)s left over: \(\frac{e^{11}}{e^5 e^4} = \boxed{e^2}\). Note that \(e\) is a fundamental constant in mathematics, like \(\pi\), equal to \(2.718281828459045\ldots\) approximately, but we just use \(e\) for the exact value.
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\(\frac{(A^4 B)^3}{(A B^4)^2}\).
We see \((A^4 B)^3 = A^{12} B^3\). On bottom, we have \((A B^4)^2 = A^2 B^8\). This give \(\frac{A^{12} B^3}{A^2 B^8}\). Once we get cancel two of the As and three of the Bs, we have \(\boxed{\frac{A^{10}}{B^5}}\).
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\(\left(\frac{\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}}{\sqrt[4]{104942}} \right)^{x - x}\).
This looks ugly, but \(x - x = 0\), and anything to the zeroth power is \(1\). Hence the answer is \(\boxed{1}\).
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\(a^{-5}a^2\).
This combines as \(a^{-5 + 2} = a^{-3}\), which we can also write as \(\boxed{\frac{1}{a^3}}\).
Fractional Exponents
One more rule before you go: . 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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.
We see this is the same thing as \(\sqrt{25}\), which is \(\boxed{5}\).
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.
This is the same thing as \(\left( \sqrt[3]{8}\right)^2\). We see that \(\sqrt[3]{8} = 2\), and hence we do \(2^2 = 4\). So .