Question

The square root of the independent term in the expansion of \[ \left( \frac{2x^2}{5} + \frac{5}{\sqrt{x}} \right)^{10} \] is:

• A. \( 15\sqrt{10} \)
• B. \( 10\sqrt{15} \)
• C. \( 30\sqrt{5} \)
• D. \( 20\sqrt{5} \)

Hint:

The independent term in a binomial expansion is found by equating the exponent of \( x \) to zero and solving for \( r \).

Answer 1

The Correct Option is C

We need to find the square root of the independent term in the expansion of: \[ \left( \frac{2x^2}{5} + \frac{5}{\sqrt{x}} \right)^{10} \] Step 1: General Term in the Binomial Expansion The general term in the binomial expansion of \( (a + b)^{n} \) is given by: \[ T_k = \binom{n}{k} a^{n-k} b^k \] For the expression \( \left( \frac{2x^2}{5} + \frac{5}{\sqrt{x}} \right)^{10} \), we identify: - \( a = \frac{2x^2}{5} \), - \( b = \frac{5}{\sqrt{x}} \), - \( n = 10 \). Thus, the general term is: \[ T_k = \binom{10}{k} \left( \frac{2x^2}{5} \right)^{10-k} \left( \frac{5}{\sqrt{x}} \right)^k \] Simplifying: \[ T_k = \binom{10}{k} \left( \frac{2^{10-k} x^{2(10-k)}}{5^{10-k}} \right) \left( \frac{5^k}{x^{k/2}} \right) \] This simplifies to: \[ T_k = \binom{10}{k} \frac{2^{10-k} 5^k}{5^{10-k}} x^{2(10-k) - k/2} \] So the exponent of \( x \) in the general term is: \[ 2(10-k) - \frac{k}{2} = 20 - 2k - \frac{k}{2} = 20 - \frac{5k}{2} \] Step 2: Finding the Independent Term For the independent term, the exponent of \( x \) must be zero. Therefore, set the exponent of \( x \) to zero: \[ 20 - \frac{5k}{2} = 0 \] Solving for \( k \): \[ \frac{5k}{2} = 20 \quad \Rightarrow \quad k = 8 \] Step 3: Finding the Value of the Independent Term Substitute \( k = 8 \) into the expression for \( T_k \): \[ T_8 = \binom{10}{8} \frac{2^{10-8} 5^8}{5^{10-8}} x^{0} \] Simplifying: \[ T_8 = \binom{10}{8} \frac{2^2 5^8}{5^2} = \binom{10}{8} \frac{4 \cdot 5^6}{25} \] Using \( \binom{10}{8} = 45 \): \[ T_8 = 45 \times \frac{4 \cdot 5^6}{25} = 45 \times \frac{4 \cdot 15625}{25} = 45 \times 2500 = 112500 \] Step 4: Finding the Square Root The square root of the independent term is: \[ \sqrt{112500} = 30 \sqrt{5} \] Thus, the correct answer is: \[ \boxed{30\sqrt{5}} \]