Chapter 7: Problem 501

6 th term in the expansion of \(\left[\left(1 / x^{(8 / 3)}\right)+x^{2} \log _{10} x\right]^{8}\) is 5600 then \(\mathrm{x}=\) (a) 2 (b) \(\sqrt{5}\) (c) \(\sqrt{(10)}\) (d) 10

Short Answer

Expert verified

Short Answer: The correct value for \(x\) to satisfy the given equation is \(x = 10\). So, the correct option is (d).

Step by step solution

Understand the expression given

The expression given is \(\left[\left(\frac{1}{x^{\frac{8}{3}}} \right) + x^2 \log_{10}{x} \right]^8\). We need to find the 6th term in its expansion and then determine the value of x.

Applying Binomial Theorem

The binomial theorem tells us how to expand expressions of the form (a+b)^n: \((a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\) Here our expression is (a+b)^8, where a = \(\frac{1}{x^{\frac{8}{3}}}\) and b = \(x^2\log_{10}x\). To find the 6th term, we can plug in k=5 since that corresponds to the 6th term.

Calculate the 6th term

When k=5, the binomial term becomes: \(\binom{8}{5}a^{8-5}b^5 = \binom{8}{5}\left(\frac{1}{x^{\frac{8}{3}}}\right)^3(x^2\log_{10}x)^5\) That simplifies to: \(56\left(\frac{1}{x^8}\right)(x^{10}\log_{10}^5x)\)

Use the given value of the 6th term

We are given that the 6th term is 5600. So, we equate the expression we found in the previous step to 5600 and solve for x: \(5600 = 56\left(\frac{1}{x^8}\right)(x^{10}\log_{10}^5x)\)

Solve for x

First, let's divide both sides by 56: \(100 = \left(\frac{1}{x^8}\right)(x^{10}\log_{10}^5x)\) Now, let's multiply both sides by x^8: \(100x^8 = x^{10}\log_{10}^5x\) Divide by \(x^2\): \(100x^6 = \log_{10}^5x\) Now, we can take the 5th root of both sides of the equation to isolate the log term: \(\log_{10}x = \sqrt[5]{100x^6}\) To get rid of the log term, we take 10 to the power of both sides of the equation: \(x = 10^{\sqrt[5]{100x^6}}\)

Test the given options for x

We now test each given option for x to see which one satisfies the equation: (a) x = 2: \(2 = 10^{\sqrt[5]{100(2)^6}}\) => not true (b) x = \(\sqrt{5}\): \(\sqrt{5} = 10^{\sqrt[5]{100(\sqrt{5})^6}}\) => not true (c) x = \(\sqrt{10}\): \(\sqrt{10} = 10^{\sqrt[5]{100(\sqrt{10})^6}}\) => not true (d) x = 10: \(10 = 10^{\sqrt[5]{100(10)^6}}\) => true Given that only option (d) satisfies the equation, the correct answer is \(x = 10\).

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6 th term in the expansion of \(\left[\left(1 / x^{(8 / 3)}\right)+x^{2} \log _{10} x\right]^{8}\) is 5600 then \(\mathrm{x}=\) (a) 2 (b) \(\sqrt{5}\) (c) \(\sqrt{(10)}\) (d) 10

Short Answer

Step by step solution

Understand the expression given

Applying Binomial Theorem

Calculate the 6th term

Use the given value of the 6th term

Solve for x

Test the given options for x

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