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Laws Exponents

Laws of Exponents

There are some properties of exponents. They are called Laws of Exponents.

Priority of exponent

Normally the exponent only refers to the immediate number under it, the base. If there is an additional coefficient, this coefficient is not concerned by the exponent. If you want to include this coefficient, take a bracket.

n = coefficient, x = base, a = exponent

The exponent refers to x.

The exponent refers both to n and x.

  Examples

The exponent 2 only belongs to x, not to 5.

The brackets show that the exponent 2 belongs both to 5 and x.

Negative exponent

A negative exponent has something to do with fraction. If you put an exponent from the denumerator to the numerator, the exponent becomes negative and vice versa. The sign changes each time you change the position of the exponent in a fraction. It is easier to manage positive exponents. So be free to arrange the position of the terms in a fraction.

The exponent was put above the fraction line. The positive sign changes to negative.

The exponent was put above the fraction line. The negative sign changes to positive.

  Examples

If you want to write a fraction in one line, you can use the sign change of the exponent.

Addition of exponents

When you multiply numbers with the same base, you can add the exponents. The base must be equal otherwise the addition of the exponents is not possible.

  Examples

Add the exponents.

For the bases are different, you can’t add the exponents.

Subtraction of exponents

When you divide numbers with the same base, you can subtract the exponents. The base must be equal otherwise the subtraction of the exponents is not possible.

  Examples

Subtract the exponents.

For the bases are different, you can’t substract the exponents.

Multiplication of exponents

A power to a power means multiply the exponents. It is important where the brackets around the base are. Normally the exponent only refers to the immediate number, not to the coefficient before this number. If you want to enclose the coefficient, take the brackets.

  Example

Exponent of 0

Zero multiplied by itself always is 0. Therefore

 .

Undefined

A zero power to zero is not defined, because .

Should  then be 0?

The zero exponent would both be 1 and 0, a contradiction!

 Exercises

1. Evaluate a³a⁴.

Add the exponents, because the base is equal.

2. Evaluate b⁷ / b^2.

Subtract the exponents, because the base is equal.

3. Evaluate 70000⁰.

70000⁰ = 1

Everything to the zero-th power is one (Exception 0⁰).

4. Evaluate 6^-2

Negative power means 1 divided by exponent number.

5. Evaluate

Change the position of a and b and subtract or add the exponents. Here you can convince yourself, how easy it is to evaluate negative exponents.