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Chapter 7: Problem 584

Numbers of rational terms in expansion \(\left.\left[3^{(1 / 2)}+5^{(1 / 8)}\right)\right]^{256}\) are (a) 33 (b) 34 (c) 35 (d) 32

Short Answer

Expert verified
The number of rational terms in the expansion \(\left[3^{\frac{1}{2}} + 5^{\frac{1}{8}}\right]^{256}\) is 33.

Step by step solution

01

Recall the binomial theorem

Recall that according to the binomial theorem, for any positive integer n and any real numbers x and y, the expansion of (x+y)^n is given by: \((x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k\) In our case, x = \(3^{\frac{1}{2}}\) and y = \(5^{\frac{1}{8}}\), with n = 256.
02

Identify rational term conditions

For a term in the expansion to be rational, both the base 3 and base 5 components need to have rational exponents. That is, \(3^{\frac{1}{2}(256-k)}\) must be a rational number, and \(5^{\frac{1}{8}k}\) must be a rational number.
03

Determine exponents required for rational numbers

Looking at the base 3 exponent: \(\frac{1}{2}(256-k) = 128 - \frac{1}{2}k\) For the expression to result in a rational number, the exponent of 3 must be an integer. Similarly, looking at the base 5 exponent: \(\frac{1}{8}k\) For the expression to result in a rational number, the exponent of 5 must be an integer.
04

Combine the conditions and solve for k

Combining the conditions, we have: \(128 - \frac{1}{2}k\) must be an integer, \(\frac{1}{8}k\) must be an integer. Now, notice that if \(\frac{1}{8}k\) must be an integer, k must be a multiple of 8. Let \(k = 8m\), where m is an integer. Now we can rewrite the base 3 exponent condition: \(128 - \frac{1}{2}(8m)\) \(128 - 4m\) This way, the integer m can take any value ranging from 0 to 32.
05

Count the number of rational terms

As m ranges from 0 to 32, we get a total of 33 values for m, and thus 33 values for k that produce rational terms in the expansion. So the number of rational terms in the given expression is 33. #Answer# (a) 33

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Most popular questions from this chapter

\(\mathrm{s}(\mathrm{k}): 1+3+5+\ldots+(2 \mathrm{k}-1)=3+\mathrm{k}^{2}\) then which statement is true ? (a) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (b) \(\mathrm{s}(\mathrm{k}) \Rightarrow \mathrm{s}(\mathrm{k}+1)\) (c) s (1) is true (d) Result is proved by Principle of Mathematical induction

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