An index is a power of numbers that indicate how many times it has been multiplied by itself. Here are examples indices: \({2^9},{x^3},x^{1/2}\) and others. In \({2^9}\), wherein the small number, 9, above is the index, it indicates that 2 will be multiplied by itself nine times or \({2^9}\) = \(2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\). The same applies to \({x^3}\) . It represents \({x^3}\) = \(x\cdot x\cdot x\). A number or variable that appears without an index example an x means x to the power of 1.

A root of a number is also an index, it represents the index that has a power of fractions. An \(\sqrt{x}\) means \({x^{1/2}}\). Another example: \(\sqrt[4]{x^3}\) is \({x^{3/4}}\).

**Rules of Indices**

**Indices in Multiplying
**In multiplying an equation, numbers or variables that have the same base and an index or power, simply add the index. Let’s have an example of \(a^2(a^3)\). Write the indices out as

\(a^2\) = \( a\cdot a\) and \(a^3\) = \(a\cdot a\cdot a\) .

\(a^2 \cdot a^3\)= \(a\cdot a\cdot a\cdot a\cdot a\) . It shows that

*a*should be multiplied by itself five times. If simplified, \(a^5\).

\(a^2\cdot a^3\) = \(a^5\)

Example 2:

\(2ab^3c\cdot3a^3c^4\) Solve it separately

\(2\cdot 3 = 6\)

\(a \cdot a^3\) = \(a^{1+3} = a^4\)

\(c\cdot c^4 = c^{1+4} = c^5\)

\(2ab^3 c\cdot 3a^3 c^4 = 6a^4 b^3 c^5\)

*Note:* Multiply if the values or variables have the same base, then add the index.

**Indices in Dividing
**In dividing an equation, numbers or variables that have the same base and an index or power, simply subtract the index. Let’s have an example \(a^4\div a^2\) . Write as the numerator and the denominator \(\frac{a^4}{a^2}\) and in full write out \(\frac{a\cdot a\cdot a \cdot a}{a\cdot a\cdot}\). Cancel out the common factor to have \(a^2\).

Example 2:

\(\frac{4ab^3c^5}{2a^2bc^3}\)

\(\frac{4}{2}=2\)

\(\frac{a}{a^2} = a^{1 -2} = a^{-1}\) A negative power will be in the denominator.

\(\frac{b^3}{b} = b^{3 – 1} = b^2\)

\(\frac{c^5}{c^3} = c^{5 – 3} = c^2\)

\(\frac{4ab^3c^5}{2a^2bc^3} = \frac{2b^2 c^2}{a}\)

*Note:* Divide a number or variable with the same base and subtract the index. In placing together, all the values with a positive power are in the numerator and negative powers are in the denominator.

**Power Raised to a Power
**An index raised to another index, simply multiply the powers together. Let’s have \(({a^4})^2\), it means that \(a^4\) is to be squared or multiplied by itself again. \(({a^4})^2 = a^4\cdot a^4 = a^{4+4} = a^8\) . Therefore, \(({a^4})^2\) is equal to \(a^8\) .

Example 2:

\({2a^5 bc^2})^4\)

\(2^4 = 16\)

\({(a^5)}^4 = a^5\cdot a^5\cdot a^5\cdot a^5 = a^{5+5+5+5} = a^20\)

\({b})^4 = b\cdot b\cdot b\cdot b = b^{1+1+1+1} = b^4\)

\(({c^2})^4 = c^2\cdot c^2\cdot c^2\cdot c^2 = c^{2+2+2+2} = c^8\)

\(({2a^5 bc^2})^4 = 16 a^{20} b^4 c^8\)

**Power of Zero
**Any number or variable raised to the power of 0 is equal to 1. Let’s have an example: \(a^0 = 1\)

Example 2:

\(({a^2})^0 = a^{2(0)} = a^0 = 1\)

Example 3:

\(\frac{b^3}{b^3} = b^{3 – 3} = b^0 = 1\)