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basics
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fractional
Arbitrary& zero
extra
100
multiplying the number of identical factors. So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ is just three factors of x, (x)(x)(x).
What Is an Exponent
100
to divide by that number of factors instead of multiplying. So 4−3 is the same as 1/(43), and x-3 = 1/x3.
What are negative exponents
100
the form 1/n — means to take the nth root instead of multiplying or dividing. For example, 4(1/3) is the 3rd root (cube root) of 4.
What is fractional exponents
100
You can’t use counting techniques on an expression like 60.1687 or 4.3π. Instead, these expressions are evaluated using
What are logarithms.
100
An exponent of 1/2 is actually square root And an exponent of 1/3 is cube root An exponent of 1/4 is 4th root And so on!
What is fractional exponents
200
the order of operations. this is the first operation (in the absence of grouping symbols like parentheses), so it applies only to what it’s directly attached to. 3x³ is 3(x)(x)(x), not (3x)(3x)(3x). If we wanted (3x)(3x)(3x), we’d need to use grouping: (3x)³.
What is the one warning about exponents
200
anything to the 0 power is 1. But now you can see why. Consider x0. By the division rule, you know that x3/x3 = x(3−3) = x0. But anything divided by itself is 1, so x3/x3 = 1. Things that are equal to the same thing are equal to each other: if x3/x3 is equal to both 1 and x0, then 1 must equal x0. Symbolically, x0 = x(3−3) = x3/x3 = 1 There’s one restriction. You saw that we had to create a fraction to figure out x0. But division by 0 is not allowed, so our evaluation works for anything to the 0 power except zero itself:
What is the zero exponent
200
Look at the problem at the right. If we subtract the exponents, we get: If we "cancel" like terms, we get:
What is negative exponents
300
Suppose you have (x5)(x6); how do you simplify that? Just remember that you’re counting factors. x5 = (x)(x)(x)(x)(x) and x6 = (x)(x)(x)(x)(x)(x) Now multiply them together: (x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x11 Why x11? Well, how many x’s are there? Five x factors from x5, and six x factors from x6, makes 11 x factors total. Can you see that whenever you multiply any two powers of the same base, you end up with a number of factors equal to the total of the two powers? In other words, when the bases are the same, you find the new power by just adding the
What are exponents
300
What about dividing by a negative power, like y5/x−4? Use the rule you already know for dividing: 5 5 5 4 5 4 y y y x y x 4 5 --- = -------- = -------- * -- = ----- = x y -4 ( 4) ( 4) 4 x (1 / x ) (1 / x ) x 1
What is negative powers on the bottom
300
a fractional exponent: the square (2nd) root of x is just x1/2, the cube (3rd) root is just x1/3, and so on. With this fact at your disposal, you’re in good shape. Example: . That’s easy to evaluate! You know that the cube (3rd) root of x is x1/3 and the square root of that is (x1/3)1/2. Then use the power-of-a-power rule to evaluate that as x(1/3)(1/2) = x(1/6), which is the 6th root of x. Example: . Why? Because the square root is the 1/2 power, and the product rule for the same power of different bases tells you that (x1/2)(y1/2) = (xy)1/2.
What are radicals
300
13. (a6b8c10 / a5b6d7)0 14. 17x0
13. 1 14. 17×1 = 17
300
(Remember that x cannot equal 0 or a division by zero error will occur.)
What is the negative exponent rule
400
(x³)(y³) = (x)(x)(x)(y)(y)(y)
What are powers of different bases
400
Write each of these as a single positive power. (I’ve slipped in one or two that can’t be simplified, just to keep you on your toes.) 5. a7 ÷ b7 6. 11² × 2³ 7. 8³ x³ 8. 54 × 56 9. p11 ÷ p6 10. r-11 ÷ r-2
5. (a/b)7 6. cannot be simplified. 7. (8x)³ 8. 510 (not 2510!) 9. p5 10. r-11-(-2) = r-9 = 1/r9
400
So far we’ve looked at fractional exponents only where the top number was 1. How do you interpret x2/3, for instance? Can you see how to use the power rule? Since 2/3 = (2)(1/3), you can rewrite x2/3 = x(2)(1/3) = (x2)1/3, which is . It works the other way, too: 2/3 = (1/3)(2), so x2/3 = x(1/3)(2) = (x1/3)2 = . These are examples of the general rule: When a power and a root are involved, the top part of the fractional exponent is the power and the bottom part is the root. Suppose p and r are the same? Then you have, for instance, . But that’s the same as x5/5, and 5/5=1, so it’s the same as x1 or just x.
What are fractional & rational exponents
400
and the log-as-inverse definition: x = 6.74.4 log x = 4.4 ( log 6.7 ) = about 3.634729132 x = 103.63472... = about 4312.5 There’s nothing special about base-10 logs here. The calculation could just as well be x = 6.74.4 ln x = 4.4 ( ln 6.7 ) = about 8.369273116 x = e8.36927... = about 4312.5 This will work for any positive base and any real exponent, so for example x = ππ log x = π (log π) = about 1.561842388 x = 101.5618... = about 36.46215961 You can combine this with the multiplying numbers = adding logarithms rule to evaluate powers that are too big for your calculator. For example, what is 671217? x = 671217 log x = 217 (log 671) = about 613.3987869 Now, separate the integer and fractional parts of the logarithm. log x = about 0.3987869 + 613 x = 100.3987869 + 613 x = 100.3987869 · 10613 x = about 2.505 · 10613 For examples like this, you really do have to use base-10 logs. other wise known as the two laws needed for arbitary expoonents
What is logarithm law and exponent law
400
no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.
What is fermat's last theorem
500
What do you do with an expression like (x5)4? There’s no need to guess — work it out by counting. (x5)4 = (x5)(x5)(x5)(x5) Write this as an array: x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x) x5 = (x) (x) (x) (x) (x) How many factors of x are there? You see that there are 5 factors in each row from x5 and 4 rows from ( )4, in all 5×4=20 factors. Therefore, (x5)4 = x20
What is powers of a power
500
What is x8÷x6? Well, there are several ways to work it out. One way is to say that x8÷x6 = x8(1/x6), but using the definition of negative exponents that’s just x8(x-6). Now use the product rule (two powers of the same base) to rewrite it as x8+(-6), or x8-6, or x2. Another method is simply to go back to the definition: x8÷x6 = (xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) = (xx)(xxxxxx÷xxxxxx) = (xx)(1) = x2. However you slice it, you come to the same answer: for division with like bases you subtract exponents, just as for multiplication of like bases you add exponents:
What is dividing powers
500
15. Write √x5 as a single power. 16. Simplify ³√(a6b9) (That’s the cube root or third root of a6b9.) 17. Find the numerical value of 274/3 without using a calculator
15. x5/2 16. (a6b9)1/3 = a²b³ 17. 274/3 = (271/3)4 by the power-of-a-power law. 271/3 is the same as the cube root of 27, which is 3. (271/3)4 = 34 = 81
500
1. Write 11³ as a multiplication. 2. Write j-7 as a fraction, using only positive exponents. 3. What’s the value of 100½? 4. Evaluate −5-2 and (−5)-2.
1. 11×11×11 2. 1/j7 3. √100 = 10 4. −5-2 = −1/25 and (−5)-2 = 1/25 (Excel returns 1/25 or 0.04 for both of these, but that’s wrong.)
500
(5 x 106) (2x 103) (3x 103) _________________________ = ______5 x 10 4
What is the coefficient exponent used for