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You can read the details below. By accepting, you agree to the updated privacy policy. Thank you! View updated privacy policy We've encountered a problem, please try again. Rational ExponentsSo far, exponents have been limited to integers. In this section, we will define what rational (or fractional) exponents mean and how to work with them. All of the rules for exponents developed up to this point apply. In particular, recall the product rule for exponents. Given any rational numbers m and n, we have xm⋅xn=xm+n For example, if we have an exponent of 1/2, then the product rule for exponents implies the following: 51/2⋅51/2 =51/2+1/2=51=5 Here 51/2 is one of two equal factors of 5; hence it is a square root of 5, and we can write 51/2=5 Furthermore, we can see that 21/3 is one of three equal factors of 2. 21/3⋅ 21/3⋅21/3=21/3+1/3+1/3=23/3=21=2 Therefore, 21/3 is a cube root of 2, and we can write 21/3=23 This is true in general, given any nonzero real number a and integer n≥2, a1/n=an In other words, the denominator of a fractional exponent determines the index of an nth root. Example 1Rewrite as a radical.
Solution:
Example 2Rewrite as a radical and then simplify.
Solution:
Example 3Rewrite as a radical and then simplify.
Solution: a. (64x3 )1/3=64x33=43x33 =4x b. (−32x5y10)1/ 5=−32x5y105=(−2)5 x5(y2)55=−2xy2 Next, consider fractional exponents where the numerator is an integer other than 1. For example, consider the following: 52/3⋅52/3⋅52/3=52/3+2 /3+2/3=56/3=52 This shows that 52/3 is one of three equal factors of 52. In other words, 52/3 is a cube root of 52 and we can write: 52/3=523 In general, given any nonzero real number a where m and n are positive integers (n≥2), am/n=amn An expression with a rational exponentThe fractional exponent m/n that indicates a radical with index n and exponent m: am/n=amn. is equivalent to a radical where the denominator is the index and the numerator is the exponent. Any radical expression can be written with a rational exponent, which we call exponential formAn equivalent expression written using a rational exponent.. Radicalform Exponentialformx2 5=x2/5 Example 4Rewrite as a radical.
Solution:
Example 5Rewrite as a radical and then simplify.
Solution: We can often avoid very large integers by working with their prime factorization. a. 272/3=2723 =(33)23 Replace27with33.=363 Simplify.=32 =9 b. ( 12)5/3=(12)53Repl ace12with22⋅3.=(22⋅3)5 3Applytherulesforexponents. =210⋅353Simpl ify.=29⋅2⋅33⋅323=23⋅3⋅2⋅323 =24183 Given a radical expression, we might want to find the equivalent in exponential form. Assume all variables are positive. Example 6Rewrite using rational exponents: x35. Solution: Here the index is 5 and the power is 3. We can write x35=x3/5 Answer: x3/5 Example 7Rewrite using rational exponents: y36. Solution: Here the index is 6 and the power is 3. We can write y36=y3/6=y1/ 2 Answer: y1/2 It is important to note that the following are equivalent. am /n=amn=(an)m In other words, it does not matter if we apply the power first or the root first. For example, we can apply the power before the nth root: 272/3=2723=(33)2 3=363=32=9 Or we can apply the nth root before the power: 272/3 =(273)2=(333)2=(3)2=9 The results are the same. Example 8Rewrite as a radical and then simplify: (−8)2/3. Solution: Here the index is 3 and the power is 2. We can write (−8)2/3=(−83) 2=(−2)2=4 Answer: 4 Try this! Rewrite as a radical and then simplify: 1003 /2. Answer: 1,000 Some calculators have a caret button ^ which is used for entering exponents. If so, we can calculate approximations for radicals using it and rational exponents. For example, to calculate 2=21/2=2^(1/2)≈1.414, we make use of the parenthesis buttons and type 2^(1÷2)= To calculate 223=22/3=2^(2/3)≈1.587, we would type 2^( 2÷3)= Operations Using the Rules of ExponentsIn this section, we review all of the rules of exponents, which extend to include rational exponents. If given any rational numbers m and n, then we have
These rules allow us to perform operations with rational exponents. Example 9Simplify: 71/3⋅74/9. Solution: 71/3⋅74/9=71/3+4/9Applytheproductrulexm⋅xn=xm+n. =73/9+4/9=77/9 Answer: 77/9 Example 10Simplify: x3/2x2 /3. Solution: x3/2x2/3=x 3/2−2/3Applythequotientrulexmxn =xm−n.=x9/6−4/6=x5/6 Answer: x5/6 Example 11Simplify: (y3/4)2/3. Solution: (y 3/4)2/3=y(3/4)(2/3)Applyt hepowerrule(xm)n=xm⋅n.= y6/12Multiplytheexponentsandreduce. =y1/2 Answer: y1/2 Example 12Simplify: (81a8b12)3/4. Solution: (81a8b12)3/4=(34a8b12 )3/4Rewrite81as34.=( 34)3/4(a8)3/4(b12)3/4A pplythepowerruleforaproduct(xy)n=x nyn.=34(3/4)a8(3/4)b12( 3/4)Applythepowerruletoeachfact or.=33a6b9Simplify.= 27a6b9 Answer: 27a6b9 Example 13Simplify: (9x4)−3/2. Solution: (9x4)−3/2=1(9x4)3/2 Applythedefinitionofnegativeexponen tsx−n=1xn.=1(32x4) 3/2Write9as32andapplyth erulesofexponents.=132(3/2) x4(3/2)=133⋅x6=127x6 Answer: 127x6 Try this! Simplify: (125a1/4b6 )2/3a1/6. Answer: 25b4 Radical Expressions with Different IndicesTo apply the product or quotient rule for radicals, the indices of the radicals involved must be the same. If the indices are different, then first rewrite the radicals in exponential form and then apply the rules for exponents. Example 14Multiply: 2⋅2 3. Solution: In this example, the index of each radical factor is different. Hence the product rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents. Then apply the product rule for exponents. 2⋅23=21/2⋅21/3Equivalentsusingrationalexponents=21/2+1/3Applytheproduc truleforexponents.=25/6=256 Answer: 256 Example 15Divide: 43 25. Solution: In this example, the index of the radical in the numerator is different from the index of the radical in the denominator. Hence the quotient rule for radicals does not apply. Begin by converting the radicals into an equivalent form using rational exponents and then apply the quotient rule for exponents. 4325=2 2325=22/3 21/5Equivalentsusingrationale xponents=22/3−1/5App lythequotientruleforexponents.=27/15=2715 Answer: 2715 Example 16Simplify: 43. Solution: Here the radicand of the square root is a cube root. After rewriting this expression using rational exponents, we will see that the power rule for exponents applies. 43=223=(22/3)1/2Equivalentsusingrationalexponents =2(2/3)(1/2)Applythepowerr uleforexponents.=21/3=23 Answer: 23 Key Takeaways
Topic Exercises
Part A: Rational ExponentsExpress using rational exponents. Express in radical form. Write as a radical and then simplify. Use a calculator to approximate an answer rounded to the nearest hundredth.
Part B: Operations Using the Rules of ExponentsPerform the operations and simplify. Leave answers in exponential form.
Part C: Radical Expressions with Different IndicesPerform the operations.
Part D: Discussion BoardAnswers
What are radicals and rational exponents?Square roots are most often written using a radical sign, like this, √4 . But there is another way to represent them. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, √4 can be written as 412 4 1 2 .
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