# Indices

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## What are Indices?

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:
$$\frac{b^3}{b^3} = b^{3 – 3} = b^0 = 1$$