ExponentsExponents can make your math problems a lot easier to handle. Simply put, it is a shortcut for multiplying numbers over and over again. Look at the following multiplication problem: 8 × 8 × 8 × 8 × 8 × 8 Instead of multiplying 8 six times by itself,we can just write 8^{6} and it will mean the same thing. When reading 8^{6}, we say eight to the sixth power or eight to the power of six. In a similar way, 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 = 12 ^{10} In 12^{10}, 12 is called the base and 10 is called the exponent Other examples: 5^{3} = 5 × 5 × 5 9^{4} = 9 × 9 × 9 × 9 7^{2} = 7 × 7 6^{6} = 6 × 6 × 6 × 6 × 6 × 6 × 6 2^{8} = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 In general, y^{n} = y × y × y × y ×...× y × y (n times) In our example above, it says 7^{2} = 7 × 7 What would you say 7^{1} is equal to? For 7^{2}, you wrote down 7 twice Therefore, for 7^{1}, you will write down 7 once and of course there is not need to have a multiplication sign 7^{1} = 7 Common pitfalls to avoid when working with exponents: What is 2^{6} equals to? Is it equal to 2 × 2 × 2 × 2 × 2 × 2 ? Or is it equal to (2 × 2 × 2 × 2 × 2 × 2) ? It is equal to (2 × 2 × 2 × 2 × 2 × 2) = (2^{6}) =  64 However, (2)^{6} is a different story (2)^{6} = (2) × (2) × (2) × (2) × (2) × (2) =  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 Notice that  ×  = + So,  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + × 2^{6} + × + = + + 2 ^{6} In (2)^{6}, the exponent is even. Change it to any odd number and the answer will be negative (2)^{7} = (2) × (2) × (2) × (2) × (2) × (2)× (2) =  ×  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 × 2  ×  ×  ×  ×  ×  ×  × 2 × 2 × 2 × 2 × 2 × 2 × 2 = + × + × + ×  × 2^{7} + × + × + ×  × 2^{7} = + ×  × 2^{7} + ×  =  + ×  × 2^{7} = 2^{7} Observation: a^{n} is always negative (a)^{n} is either negative or positive. It is positive is an is an even number. It is negative if n is an odd number a^{n} is not always equal to (a)^{n} a^{n} is equal to (a)^{n} only when n is an odd number What if the base is a fraction? When the base is a fraction it is common to use to use parentheses as shown below:
The above is also equal to
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