• two examples of expanding the exponent are:$3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$ and$2^{-3} = (\frac{1}{2})^3 = \frac{1}{2^3} = \frac{1}{2 \cdot 2 \cdot 2} = \frac{1}{8}$
• a root of a number involves a fractional exponent for example the quantity$x^{\frac{1}{n}}$ is the nth root of$x$
• when multiplying powers of the same base add exponents togethor:$x^a \cdot x^b = x^{a+b}$
• when dividing powers of the same base you subtract the exponents:$\frac{x^a}{x^b} = x^a \div x^b = x^{a-b}$
• any number except zero when raised to the power of zero equals one:$n^3 \div n^3 = n^{3-3} = n^0 = 1$ as long as$n \neq 0$
• the power of power is:$(a^m)^p = a^{m \cdot p}$ for example$(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$
• when raising two numbers to the same power their products and quotients obey simple rules:$(x^n)(y^n) = (xy)^n$ and$\frac{x^n}{y^n} = (\frac{x}{y})^n$

• the equation to convert celsius to fahrenheit is:$F = C \cdot \frac{9}{5} + 32$
• the equation of a unit circle is:$x^2 + y^2 = 1$
• the relationship between sine and cosine is famously shown with the Pythagorean identity:$\sin^2\theta + \cos^2\theta = 1$
• $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• absolute value or modulus of a complex number$a + bi$ is$$|a + bi| = \sqrt{a^2 + b^2}$$
• in Boyle's law the product of pressure (P) and volume (V) is a constant number (k):$P \cdot V = k$ which means that pressure is inversely proportional to the volume

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