Powers and exponents

We know how to calculate the expression 5 x 5. This expression can be written in a shorter way using something called exponents.

$$5\cdot 5=5^{2}$$

An expression that represents repeated multiplication of the same factor is called a power.

The number 5 is called the base, and the number 2 is called the exponent. The exponent corresponds to the number of times the base is used as a factor.

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Example

Write these multiplications like exponents

$$5\cdot 5\cdot 5=5^{3}$$

$$4\cdot 4\cdot 4\cdot 4\cdot 4=4^{5}$$

$$3\cdot 3\cdot 3\cdot 3=3^{4}$$

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Multiplication

If two powers have the same base then we can multiply the powers. When we multiply two powers we add their exponents.

The rule:

$$x^{a}\cdot x^{b}=x^{a+b}$$

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Example

$$4^{2}\cdot 4^{5}=\left ( 4\cdot 4 \right )\cdot \left ( 4\cdot 4\cdot 4\cdot 4\cdot 4 \right )=4^{7}=4^{2+5}$$

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Division

If two powers have the same base then we can divide the powers. When we divide powers we subtract their exponents.

The rule:

$$\frac{x^{a}}{ x^{b}}=x^{a-b}$$

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Example

$$\frac{4^{2}}{ 4^{5}}=\frac{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}}{{\color{red} {\not}{4}}\cdot {\color{red} {\not}{4}}\cdot 4\cdot 4\cdot 4}=\frac{1}{4^{3}}=4^{-3}=4^{2-5}$$

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A negative exponent is the same as the reciprocal of the positive exponent.

$$x^{-a}=\frac{1}{x^{a}}$$

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Example

$$2^{-3}=\frac{1}{2^{3}}$$

When you raise a product to a power you raise each factor with a power

$$(x\cdot y)^{a}=x^{a}\cdot y^{a}$$

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Example

$$(2x)^{4}=2^{4}\cdot x^{4}=16x^{4}$$

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The rule for the power of a power and the power of a product can be combined into the following rule:

$$(x^{a}\cdot y^{b})^{z}=x^{a\cdot z}\cdot y^{b\cdot z}$$

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Example

$$(x^{3}\cdot y^{4})^{2}=x^{3\cdot 2}\cdot y^{4\cdot 2}=x^{6}\cdot y^{8}$$

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Video lessons

Rewrite the expressions

$$2\cdot 2\cdot 2$$

$$x\cdot x\cdot x\cdot x\cdot x$$

$$3^{4}$$

$$x^{3}$$

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Simplify the expression

$$\left ( x^{2}\cdot y^{3}\cdot z^{5} \right )^{3}$$