In the expansion of X the power of seven parentheses X0 4/2+ K over X to the power of three parentheses to the power of seven the constant term is 168 find the value of K

Let's break down the given expression step by step.

We are asked to find the value of \( K \) in the expansion of:

\[
\left( x^7 \left( \frac{x^0 4}{2} + \frac{K}{x^3} \right) \right)^7
\]

Let's clarify the expression first. The original expression seems unclear with the notation provided. Based on interpretation, I assume the correct expression is:

\[
\left( x^7 \left( \frac{4}{2} + \frac{K}{x^3} \right) \right)^7
\]

Simplifying the inner terms:

\[
\left( x^7 \left( 2 + \frac{K}{x^3} \right) \right)^7
\]

Now, we distribute the \( x^7 \) inside the parentheses:

\[
\left( x^7 \cdot 2 + x^7 \cdot \frac{K}{x^3} \right)^7 = \left( 2x^7 + Kx^4 \right)^7
\]

We now expand \( \left( 2x^7 + Kx^4 \right)^7 \) using the binomial theorem:

\[
(2x^7 + Kx^4)^7 = \sum_{r=0}^{7} \binom{7}{r} (2x^7)^{7-r} (Kx^4)^r
\]

We want the constant term in the expansion. To find that, we look for the term where the powers of \( x \) cancel out, i.e., where the exponent of \( x \) is zero.

### General term in the expansion:
\[
\binom{7}{r} (2x^7)^{7-r} (Kx^4)^r = \binom{7}{r} 2^{7-r} x^{7(7-r)} K^r x^{4r}
\]
\[
= \binom{7}{r} 2^{7-r} K^r x^{49 - 3r}
\]

For the constant term, the exponent of \( x \) must be zero:
\[
49 - 3r = 0
\]
Solving for \( r \):
\[
r = \frac{49}{3} = 7
\]

So, we need the term with \( r = 7 \).

Substituting \( r = 7 \) into the general term:
\[
\binom{7}{7} 2^{7-7} K^7 x^{49 - 3(7)} = \binom{7}{7} K^7 x^0
\]

Thus, the constant term is \( K^7 \).

We are given that the constant term is 168, so:
\[
K^7 = 168
\]

Taking the seventh root of both sides:
\[
K = \sqrt[7]{168}
\]

Let me calculate the exact value of \( K \).The value of \( K \) is approximately \( 2.08 \).

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Discover how to find the value of K in the expansion of (2x^7 + Kx^4)^7 where the constant term is 168. This problem involves binomial expansion, exponent laws, and solving for constants. Suitable for high school students in Grades 10-12.

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