3.1: Algebra Tips and Tricks Part V (Exponents)

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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:

$\frac{e^{11}}{e^5 e^4}$.

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.
$\frac{(A^4 B)^3}{(A B^4)^2}$.

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 $\boxed{\frac{A^{10}}{B^5}}$.
$\left(\frac{\sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{7}}{\sqrt[4]{104942}} \right)^{x - x}$.

This looks ugly, but $x - x = 0$, and anything to the zeroth power is $1$. Hence the answer is $\boxed{1}$.
$a^{-5}a^2$.

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:

$quicklatex.com-d7660a2021fbfd293d162356e9c20fc4_l3.png$ .

We see this is the same thing as $\sqrt{25}$, which is $\boxed{5}$.
$quicklatex.com-34aa33010b07b19128fe95b2dda23615_l3.png$ .

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 .