# Exponent rules

There are many properties and rules of exponents that can be used to simplify algebraic equations. Below are some of the most commonly used. Note that the terms "exponent" and "power" are often used interchangeably to refer to the superscripts in an expression. For example, in the term Qb^{n}, Q is the coefficient, b is the base, and n is the exponent or power, as shown in the figure below.

## Addition and subtraction

The addition and subtraction of exponents are governed by the same rules.

### Adding exponents with the same base

To add or subtract terms that contain exponents, the terms must have the same base and the same power. Otherwise, the terms cannot be added. If the base and power are the same, then the coefficients of the bases can be added or subtracted, while keeping the base and power the same. Given that P and Q are constant coefficients, this can be expressed as:

Examples

1. 3(3^{2}) + 3^{2}:

3(3^{2}) + 3^{2} = (3 + 1)(3^{2}) = 4(3^{2}) = 36

2. 3x^{5} - 6x^{5}:

3x^{5} - 6x^{5} = (3 - 6)x^{5} = -3x^{5}

### Adding exponents with different powers

Recall that when working with terms containing exponents, the terms can only be added if the base and exponent of each term are the same. In cases where either the base or exponent differ between terms, the terms cannot be combined, and must be computed separately:

Examples

1. 3^{2} + 3^{3}:

In this example, while the base in each term is the same, the exponents differ. Thus, the terms cannot be combined and must be computed separately:

3^{2} + 3^{3} = 9 + 27 = 36

2. 2^{2} + 3^{2}:

2^{2} + 3^{2} = 4 + 9 = 13

In this case, the bases are the same but the exponents are not, so the terms cannot be directly combined and must be computed separately prior to addition.

## Multiplication

To multiply terms containing exponents, the terms must have the same base and/or the same power. If the exponents have coefficients attached to their bases, multiply the coefficients together. Coefficients can be multiplied together even if the exponents have different bases.

### Multiplying exponents with the same base

To multiply terms with the same base, keep the same base and add the powers together:

### Multiplying exponents with different bases

To multiply terms with different bases but the same power, raise the product of the bases to the power. This can be expressed as:

Below are some examples of multiplying exponents with the same base, different base, and same power and base.

Examples

1. 3^{2} × 3^{3}:

3^{2} × 3^{3} = 3^{2+3} = 3^{5}

2. 4^{2} × 6^{2}:

4^{2} × 6^{2} = (4 × 6)^{2} = 24^{2} = 576

If exponents have the same power and the same base, the expression can be simplified using either of the above rules:

3. 5^{2} × 5^{2}:

5^{2} × 5^{2} = 5^{2+2} = 5^{4} = 625

OR

5^{2} × 5^{2} = (5 * 5)^{2} = 25^{2} = 625

## Division

To divide terms in an expression with exponents, the exponents must have the same base and/or the same power. To divide exponents that have the same base, keep the same base and subtract the power of the denominator from the power of the numerator. If the terms of an expression have the same power but different bases, divide the bases then raise the result to the power. If the exponents have coefficients attached to their bases, divide the coefficients. Coefficients can be divided even if the exponents have different bases.

Examples

1. :

2. :

If an exponent has a negative power, you still need to keep the same sign and subtract the power.

3. :

## Negative exponents

A negative exponent just means to take the reciprocal of the base, then raise it to the positive power. This can be written as:

Example

## Powers

When raising a power to another power, it is important to pay attention to order of operations. By convention:

This is different from (b^{n})^{m}, where:

1. (4^{3})^{2}:

(4^{3})^{2} = 4^{3×2} = 4^{6} = 4096

OR

(4^{3})^{2} = 64^{2} = 4096

2. 4^{32}:

4^{32} = 4^{(32)} = 4^{9} = 262144