# Indices

Indices refers to how many times a number or letter has been multiplied by itself

1. Example of an Indice

${B}^{P}=a$

2. Indices Interpretation

B: is the Base

P: is the Power

## Laws of Indices

1. ${a}^{0}=1$

2. ${a}^{-m}=\frac{1}{{a}^{m}}$

3. ${a}^{m}×{a}^{n}={a}^{m + n}$

4. ${a}^{m}÷{a}^{n}={a}^{m - n}$

5.${\left({a}^{m}\right)}^{n}={a}^{m n}$

6. ${a}^{m / n}=\sqrt[n]{{a}^{m}}$

## Laws of Indices Explained

1.) The first law indicates that a base raised to the power 0 (Zero) is 1 (one)

2.) The second law indicates that when the power is a negative, 1 (one) becomes a numerator which is divided by the base and power.

3.) Third law indicates that when the indices are multiplying each other the powers are added together

4.) The Fourth law indicates that when the indices are dividing each other the powers are always subtracted from each other.

5.) When an indice has another power outside the brackets, the two powers multiply each other to simplify those indices.

6.) The last law of indices indicates that when powers are dividing each other and are sharing the same base, the power which is a denominator will be the root function and the other power which is a numerator will be the power for the base inside the root function.

### Point to Note:

When the bases are the same between the two indices the powers raised are used to solve the equation which maybe for x or any variable

## Example of Indices

Solve the following equation

${5}^{2y + 1}={\mathrm{25}}^{3y}$

Solution:

1.  ${5}^{2y + 1}={\mathrm{25}}^{3y}$

- Make the bases equal

2.  ${5}^{2y + 1}={5}^{2\left(3y\right)}$

- Make an equation from the powers raised

3.  $2y + 1=\mathrm{6y}$

- arrange like terms together

4.  $2y - 6y=\mathrm{-1}$

- simplify the equation and find the value of y

5.   $-4y=\mathrm{-1}$

$\frac{\mathrm{-4y}}{-4}=\frac{\mathrm{-1}}{-4}$

Answer: y = $\frac{1}{4}$

## How to solve an indice step by step

Solve the following indice

${\mathrm{100}}^{\frac{\mathrm{-1}}{2}}$

Solution:

1.   ${\mathrm{100}}^{\frac{\mathrm{-1}}{2}}$

- Make the raised fraction power 2 has the square root

2.  $\sqrt[]{{\mathrm{100}}^{-1}}$

- Find the square root of 100

3.  ${\mathrm{10}}^{-1}$

- simplify to find the answer

$\frac{1}{10}$

Answer: = $\frac{1}{10}$

## How to Simplify the indices

Find the value of an indice

${5}^{3}×{5}^{-1}×{8}^{0}$

Solution:

1.   ${5}^{3}×{5}^{-1}×{8}^{0}$

- Simplify the indices with the same bases by adding the powers

2.   ${5}^{3 - 1}×{8}^{0}$

- simplify the indice with the base having the power raised to 0 into 1

3.  ${5}^{2}×1$

- simplify to find the answer

Answer: = $25$

## Evaluating indices

Evaluate ${\left(\frac{\mathrm{64}}{125}\right)}^{\frac{\mathrm{-1}}{3}}$

Solution:

$\sqrt[3]{{\left(\frac{\mathrm{64}}{125}\right)}^{-1}}$

${\left(\frac{4}{5}\right)}^{-1}$

$\frac{1}{{\left(\frac{4}{5}\right)}^{1}}$

$\frac{1}{\frac{4}{5}}$

$\frac{1}{1}÷\frac{4}{5}$

$\frac{1}{1}×\frac{5}{4}$

$\frac{5}{4}$

Answer: $\frac{5}{4}$ or $1\frac{1}{4}$

## Evaluating indices with a power

Evaluate $\sqrt[3]{{\left(81\right)}^{3}}$

Solution:

$\sqrt[3]{{\left(81\right)}^{3}}$

${\left(3\right)}^{3}$

27

## Find the value of an indices

Find the value of x

${8}^{2x + 6}=512$

Solution:

${8}^{2x + 6}=512$

${8}^{2x + 6}={8}^{2}$

$2x + 6=2$

$2x=\mathrm{2 - 6}$

$2x=\mathrm{-4}$

$\frac{\mathrm{2x}}{2}=\frac{\mathrm{-4}}{2}$