### Power of Numbers

Welcome visitors..

In this post, we will explain about power of numbers.

The properties of exponents of numbers are as follows:

$$* \quad a^b.a^c=a^{b+c}$$ $$* \quad \frac{a^b}{a^c}=a^{b-c}$$ $$* \quad a^m.b^m=(a.b)^m$$ $$* \quad (a^b)^c=a^{b.c}$$ $$* \quad a^{\frac{b}{c}}=\sqrt[c]{a^b}$$

$$* \quad a^b.a^c=a^{b+c}$$ $$* \quad \frac{a^b}{a^c}=a^{b-c}$$ $$* \quad a^m.b^m=(a.b)^m$$ $$* \quad (a^b)^c=a^{b.c}$$ $$* \quad a^{\frac{b}{c}}=\sqrt[c]{a^b}$$

**Example 1:**

Prove that $a^0=1$

**Solution:**

From nature $\displaystyle \frac{a^b}{a^c}=a^{b-c}$, we take the value $b=c=d$ so that $$\frac{a^b}{a^c}=a^{b-c}$$ $$\frac{a^d}{a^d}=a^{d-d}$$ $$1=a^0$$ (proven).

**Example 2:**

Prove that $\displaystyle \sqrt{a}.\sqrt{b}=\sqrt{a.b}$.

**Solution:**

From nature $\displaystyle a^{\frac{b}{c}}=\sqrt[c]{a^b}$ and $\displaystyle a^m.b^m=(a.b)^m$ so that $$\sqrt{a}.\sqrt{b}=a^{\frac{1}{2}}.b^{\frac{1}{2}}$$ $$=(a.b)^{\frac{1}{2}}=\sqrt{a.b}$$ (proven).

This is an explanation of the power of numbers. Bye and thank you.