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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.

figure10

 

31

3 to the first power

3

 

42

4 to the second power or 4 squared

4 ∙ 4

 

53

5 to the third power or 5 cubed

5 ∙ 5 ∙ 5

 

26

2 to the power of six

2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

 

 

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} \\

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}\\

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}\\

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} \\

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} \\

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

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

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} \\

Example:

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

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}\\

Example:

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

Video lesson: Rewrite the expressions

\\2\cdot 2\cdot 2 \\x\cdot x\cdot x\cdot x\cdot x \\3^{4} \\x^{3}

Video lesson: Simplify the expression

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

Next Class:  Discover fractions and factors, Multiplying polynomials and binomials