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the fourth term in the expansion of $(ax + \sqrt{2}) ^ {10}$ is $30x ^ 7 10$

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Pascal's triangle and the Binomial Theorem

Full solution

Q. the fourth term in the expansion of

(ax + \sqrt{2}) ^ {10}

is

30x ^ 7 10

Use Binomial Theorem: To find the fourth term in the expansion of $(ax + \sqrt{2})^{10}$ , we use the binomial theorem. The general term is given by $T(k+1) = C(n, k) \cdot (ax)^{n-k} \cdot (\sqrt{2})^k$ , where $n=10$ and $k$ is the term number minus $1$ .
Calculate Fourth Term: For the fourth term, $k=3$ , so we plug in the values: $T(4) = C(10, 3) \cdot (ax)^{(10-3)} \cdot (\sqrt{2})^3$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $C(10, 3) = \frac{10!}{3! \cdot (10-3)!} = \frac{10\cdot9\cdot8}{3\cdot2\cdot1} = 120$ .
Plug in Values: Now plug in the values: $T(4) = 120 \times (ax)^7 \times (\sqrt{2})^3$ .
Simplify the Term: Simplify the term: $T(4) = 120 \times a^7 \times x^7 \times 2^{\frac{3}{2}} = 120 \times a^7 \times x^7 \times 2 \times \sqrt{2}$ .
Set Equal and Solve: We know that $T(4) = 30x^7$ , so we set them equal: $120 \cdot a^7 \cdot x^7 \cdot 2 \cdot \sqrt{2} = 30x^7$ .
Isolate Coefficient: Divide both sides by $x^7$ to isolate the coefficient: $120 \times a^7 \times 2 \times \sqrt{2} = 30$ .
Simplify the Equation: Simplify the equation: $240 \times a^7 \times \sqrt{2} = 30$ .
Solve for $a^7$ : Divide both sides by $240 \times \sqrt{2}$ to solve for $a^7$ : $a^7 = \frac{30}{240 \times \sqrt{2}}$ .
Find Value of 'a': Simplify the right side: $a^7 = \frac{30}{240 \cdot \sqrt{2}} = \frac{1}{8 \cdot \sqrt{2}}$ .
Find Value of 'a': Simplify the right side: $a^7 = \frac{30}{240 \sqrt{2}} = \frac{1}{8 \sqrt{2}}$ .Take the $7$ th root of both sides to solve for 'a': $a = \left(\frac{1}{8 \sqrt{2}}\right)^{\frac{1}{7}}$ .

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