Introduction to Exponents
I was x years old in the year x^{2}.
—Augustus De Morgan (1806–1871) British Mathematician
In this lesson, you'll learn how to work with terms that have positive and negative exponents, and we'll introduce fractional exponents.
Previously you learned that you can only add and subtract terms that have like bases and exponents. You also learned how to multiply terms. After multiplying the coefficients, if two terms have the same base, you added their exponents. In that same lesson, you learned that when dividing terms, if the terms had the same base, you subtract the exponent of the divisor from the exponent of the dividend. What else can we do with exponents? For starters, we can raise a base with an exponent to another exponent.
You already know that x^{4} means (x)(x)(x)(x). But what does (x^{4})^{2} mean? The term x^{4} is raised to the second power. When a term is raised to the second power, or squared, we multiply it by itself: (x^{4})^{2} = (x^{4})(x^{4}). Now we have a multiplication problem. Keep the base and add the exponents: 4 + 4 = 8, so (x^{4})^{2} = (x^{4})(x^{4}) = x^{8}.
What does (x^{3})^{6} mean? It means multiply x^{3} six times: (x^{3})(x^{3})(x^{3})(x^{3})(x^{3})(x^{3}). Adding the exponents, we can see that (x^{3})^{6} = x^{18}. Is there an easier way to find (x^{3})^{6}, without having to write out a long multiplication sentence? Yes. When raising a base with an exponent to another exponent, multiply the exponents.
Look again at (x^{3})^{6}: (3)(6) = 18, and (x^{3})^{6} = x^{18}.
Example 
What is (x^{5})^{8}? 
Multiply the exponents: (5)(8) = 40, so (x^{5})^{8} = x^{40}. 
If the term has a coefficient, raise the coefficient to the exponent, too. To find (4k^{2})^{7}, find (47)(k^{2})^{7}: (47) = 16,384. (2)(7) = 14, so (k^{2})^{7} = k^{14}. (4k^{2})^{7} = 16,384k^{14}.
Be sure to look carefully at a term before multiplying exponents. In the expression (5x^{5})^{8}, the entire term, coefficient and base, is raised to the eighth power. In the expression 6(a^{4})^{5}, only the base and its exponent, a^{4}, is raised to the fifth power. Why? Remember the order of operations. In (5x^{5})^{8}, 5x^{5} is in parentheses, which means do multiplication in parentheses before the term is raised to the eighth power. Because we cannot simplify 5x^{5} any further, raise the entire term to the eighth power. In the expression 6(a^{4})^{5}, the 6 is outside the parentheses. Exponents come before multiplication in the order of operations, so raise a^{4} to the fifth power before multiplying by 6.
Example 
What is 3(2m^{2})^{5}? 
The term 2m^{2} is raised to the fifth power and multiplied by 3. Because exponents come before multiplication in the order of operations, begin by raising 2m^{2} to the fifth power: 25 = 32 and (m^{2})^{5} = m^{10}, so (2m^{2})^{5} = 32m^{10}. Now, multiply by 3: 3(32m^{10}) = 96m^{10}. 
If a term contains more than one variable raised to an exponent, each variable must be raised to the exponent. To find (a^{2}b^{2})^{6}, raise both a^{2} and b^{2} to the sixth power. (a^{2})^{6} = a^{12} and (b^{2})^{6} = b^{12}, so (a^{2}b^{2})^{6} = a^{12}b^{12}.
Example 
What is 12( f^{3}g^{2}h^{7})^{4}? 
Raise each part within the parentheses to the fourth power: ( f^{3})^{4} = f^{12}, 
(g^{2})^{4} = g^{8}, and (h^{7})^{4} = h^{28}. ( f^{3}g^{2}h^{7})^{4} = f^{12}g^{8}h^{28}. 
Now, multiply by 12: (12)( f^{12}g^{8}h^{28}) = 12f^{12}g^{8}h^{28}. 

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