# Exponentiation

The power notation represents a multiplication of the same number. For example, 2 · 2 · 2 · 2 · 2 = 2^{5}

The number that is multiplied is called the base number and the number that expresses the number of multiplications is the exponent. In the notation 2^{5}, 2 is the base number and 5 is the exponent (or power).

3^{4} = 3 · 3 · 3 · 3 base 3, exponent 4

4^{2} = 4 · 4 base 4, exponent 2

The product of exponents with the same base.

If we expand 2^{2}·2^{3} , we will have 2^{2} ·2^{3} = 2·2·2·2·2 = 2^{5}

The short way to do this with same base number is to add the exponents 2^{x} ·2^{y} = 2^{x+y}

The quotient of exponents with the same base.

2^{4} = 2·2·2·2

2^{2} = 2·2

If we expand both exponents, we can reduce the fraction and the final answer is 2^{2}

If exponentiation with same base are divided, the exponents are subtracted from each other

Raising the exponentiation by a power.

(2^{3})^{2} We can expand the exponentiation notation (2^{3})^{2} = 2^{3} ·2^{3} Product with same base (2^{3})^{2} = 2^{3} ·2^{3} = 2^{3+3} = 2^{6}

The rule is that when the power is raised by a power, the exponents are multiplied. (2^{x})^{y} = 2^{x·y}

(2^{3})^{2} = 2^{3·2} = 2^{6}

Raising the product by a power.

(2a)^{2}

We can expand the exponentiation notation (2a)^{2} = 2a·2a

The order of multiplication doesn't matter

2 · 3 = 3 · 2 and 2·3·4 = 4·2·3.

So we rearrange (2a)^{2} = 2a·2a = 2·2·a·a = 2^{2} ·a^{2} = 4a^{2}.

All factors from the product must be raised by the power, so (a·b)^{2} = a^{2}·b^{2}

Raising the quotient by a power.

When the quotient is raised by a power, both the numerator and the denominator are raised by that power.

Negative exponents and an exponent of 0

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