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# Exponents
\\( x^n = x_1 \times x_2 \times\ \cdot\cdot\cdot\ \times x_n \\)
## Examples
* \\(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \\)
* \\(5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125 \\)
# Exponent Operations
## Multiplication
If an exponented number is multiplied by another exponented number, the
exponents would be summed together (instead of being multiplied)
\\( a^x \times b^y = (a \times b)^{x + y} \\)
* \\( 10^4 \times 2^2 = (10 \times 2)^{4 + 2} = 20^6 \\)
## Division
If an exponented number is multiplied by another exponented number, the
exponents would be subtracted together (instead of being divided)
\\( a^x \div b^y = (a \div b)^{x - y} \\)
* \\( 10^3 \div 2^4 = (10 \div 2)^{(4 - 2)} = 5^2 = 25 \\)
## Exponent
If an exponented number is exponented, multiply the exponents together.
\\( (a^x)^y = a^{x \times y} \\)
* \\( (2^2)^3 = 2^{2 \times 3} = 2^6 = 64 \\)
# Special Exponents
Some exponents feel better than the other, and Mathemathics agrees with them.
## Zero Exponent
If a number is raised to the power of **zero**, the result is **one**.
\\( a^0 = 1 \\)
* \\( 5^0 = 1 \\)
* \\( 69.420^0 = 1 \\)
## Negative Exponent
If a number is raised to a negative power, also called as an inverse, it would
be equal to...
\\( a^{-b} = \dfrac{1}{a^b} \\)
## Fractional Exponent
**Denominator (y)** root of the **number (a)** to the power of the **Numerator (x)**.
\\( a^{\frac{x}{y}} = \sqrt[y]{a^x} \\)
* \\( 5^{\frac{1}{2}} = \sqrt[2]{5} \\)
* \\( 7^{\frac{2}{3}} = \sqrt[3]{7^2} \\)
# Exponential Equation
1. $$
\begin{aligned}
2^{x + 5} &= 64 \\\\
2^{x + 5} &= 2^6 &\leftarrow\textnormal{ you can change 64 into an exponented two (}2^x\textnormal{)} \\\\
x + 5 &= 6 &\textnormal{eliminate the 2s} \\\\
x &= 6 - 5 \\\\
&= 1
\end{aligned}
$$
1. $$
\begin{aligned}
16^{x + 2} &= 2^5 \\\\
(2^4)^{x + 2} &= 2^5 \\\\
2^{4x + 8} &= 2^5 \\\\
4x + 8 &= 5 \\\\
4x &= 5 - 8 \\\\
4x &= -3 \\\\
x &= -\dfrac{3}{4}
\end{aligned}
$$
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