Exponents Study Guide (page 2)
Introduction to Exponents
I was x years old in the year x2.
—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 x4 means (x)(x)(x)(x). But what does (x4)2 mean? The term x4 is raised to the second power. When a term is raised to the second power, or squared, we multiply it by itself: (x4)2 = (x4)(x4). Now we have a multiplication problem. Keep the base and add the exponents: 4 + 4 = 8, so (x4)2 = (x4)(x4) = x8.
What does (x3)6 mean? It means multiply x3 six times: (x3)(x3)(x3)(x3)(x3)(x3). Adding the exponents, we can see that (x3)6 = x18. Is there an easier way to find (x3)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 (x3)6: (3)(6) = 18, and (x3)6 = x18.
|What is (x5)8?|
|Multiply the exponents: (5)(8) = 40, so (x5)8 = x40.|
If the term has a coefficient, raise the coefficient to the exponent, too. To find (4k2)7, find (47)(k2)7: (47) = 16,384. (2)(7) = 14, so (k2)7 = k14. (4k2)7 = 16,384k14.
Be sure to look carefully at a term before multiplying exponents. In the expression (5x5)8, the entire term, coefficient and base, is raised to the eighth power. In the expression 6(a4)5, only the base and its exponent, a4, is raised to the fifth power. Why? Remember the order of operations. In (5x5)8, 5x5 is in parentheses, which means do multiplication in parentheses before the term is raised to the eighth power. Because we cannot simplify 5x5 any further, raise the entire term to the eighth power. In the expression 6(a4)5, the 6 is outside the parentheses. Exponents come before multiplication in the order of operations, so raise a4 to the fifth power before multiplying by 6.
|What is 3(2m2)5?|
|The term 2m2 is raised to the fifth power and multiplied by 3. Because exponents come before multiplication in the order of operations, begin by raising 2m2 to the fifth power: 25 = 32 and (m2)5 = m10, so (2m2)5 = 32m10. Now, multiply by 3: 3(32m10) = 96m10.|
If a term contains more than one variable raised to an exponent, each variable must be raised to the exponent. To find (a2b2)6, raise both a2 and b2 to the sixth power. (a2)6 = a12 and (b2)6 = b12, so (a2b2)6 = a12b12.
|What is 12( f3g2h7)4?|
|Raise each part within the parentheses to the fourth power: ( f3)4 = f12,|
|(g2)4 = g8, and (h7)4 = h28. ( f3g2h7)4 = f12g8h28.|
|Now, multiply by 12: (12)( f12g8h28) = 12f12g8h28.|
So far, all of the exponents we have seen in this lesson have been positive. But we can work with negative exponents, too. Remember: a positive number multiplied by a negative number will give you a negative product, so a positive exponent multiplied by a negative exponent will give you a negative exponent:(x3)–5 = x–15, because (3)(–5) = –15.
|What is (d–3)–9?|
|A negative number multiplied by a negative number will give you a positive product: (–3)( –9) = 27, so (d–3)–9 = d27.|
Previously, we learned that bases with negative exponents appear as positive exponents in the denominator when the term is expressed as a fraction. When a coefficient is raised to a negative exponent, express the result as a fraction. Make the numerator of the fraction 1, and place the base in the denominator with a positive exponent:4–2 is equal to , which is .
|What is (5m3)–4?|
|5m3 is in parentheses, so the entire term is raised to the negative fourth power: (3)(–4) = –12, so (m3)–4 = m–12. (5)–4 = , which is .|
|(5m3)–4 = , which could also be written as .|
An expression with a term that has an exponent of 0 is the easiest to simplify. Any value—number or variable—raised to the power of 0 is 1: 10 = 1, 20 = 2, 1000 = 1, x0 = 1, and (abcd)0 = 1. We must still remember our order of operations though. The term 3x0 has only x raised to the power of zero, so this term is equal to 3(x0) = 3(1) = 3. Only the base that is raised to the zero power is equal to 1. If a term has a coefficient or another base that is not raised to the zero power, those parts of the term must still be evaluated.
|What is 4a0b2?|
|a0 = 1, so 4(a0)(b2) = 4(1)(b2) = 4b2|
A constant or variable can also be raised to a fractional exponent. The numerator of the exponent is the power to which the constant or variable is raised. The denominator of the exponent is the root that must be taken of the constant or variable. For instance x, means that x is raised to the third power, and then the second, or square, root of x must be taken. The root symbol, or radical symbol, looks like this: √. The square root of x is √x. To show that x is raised to the third power, we put the exponent inside the radical: √x3, therefore, = √x3.
If we needed to find a larger root of x, such as 5 or 6, that number would be placed just outside the radical symbol: x is the fifth root of x2 raised to the second power. We place x2 inside the radical and 5 outside the radical: x = 5√ x2.
Find practice problems and solutions for these concepts at Exponents Practice Questions.
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