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# Basic laws of calculating exponents and examples

Get to know the math exponents, their basic properties and review some examples of using that.

First of all, there is a table of basic exponents. With these properties you can solve many tasks included rational exponents.

\begin{array}{|c|c|} \hline a^n\cdot a^m=a^{n+m} & a^n:a^m=a^{n-m} \\\hline a^0=1 & (a^n)^m=a^{n\cdot m} \\\hline (a\cdot b)^n=a^n\cdot b^n & \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \\\hline a^{-n}=\frac{1}{a^n} & \left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n \\\hline \end{array}

## Solved examples

Let's solve some example tasks to see how easy to find all values of these terms.

Solve this expression $\displaystyle 2^{-2}\cdot 5^{0}\cdot 4^{2}:3^{-3}$

$$2^{-2}\cdot 5^{0}\cdot 4^{2}:3^{-3}=\frac{1}{2^2}\cdot 1 \cdot 16:\frac{1}{3^3}=108$$

Simplify this expression $\displaystyle x^{8}\cdot x^{-2}\cdot x^{-4}:x^{-3}$

$$x^{8}\cdot x^{-2}\cdot x^{-4}:x^{-3}=x^{8+(-2)+(-4)-(-3)}=x^{5}$$

Simplify this expression $\displaystyle 2^{-20}\cdot 3^{15}\cdot 2^{30}:3^{5}$

$$2^{-20}\cdot 3^{15}\cdot 2^{30}:3^{5}=2^{-20+30}\cdot 3^{15-5}=2^{10}\cdot 3^{10}=6^{10}$$

Simplify this expression $\displaystyle\frac{105(x^{-2}y^{-4})^{-2}}{3(x^3)^{-2}(y^7)^{-3}}$

$$\frac{105(x^{-2}y^{-4})^{-2}}{3(x^3)^{-2}(y^7)^{-3}}=\frac{105x^{4}y^{8}}{3x^{-6}y^{-21}}=35x^{10}y^{29}$$

Simplify this expression $\displaystyle\frac{8^{-20}\cdot 128^{5}}{16^{-10}\cdot 4^{30}}$

$$\frac{8^{20}\cdot 128^{5}}{16^{10}\cdot 4^{30}}=\frac{(2^3)^{20}\cdot (2^7)^{5}}{(2^4)^{10}\cdot (2^2)^{30}}=\frac{2^{60}\cdot 2^{35}}{2^{40}\cdot 2^{60}}=\frac{2^{95}}{2^{100}}=2^{-5}$$