Third term in expression of `(x + x^(log_(10)x))^(5)` is `10^(6)` than possible value of `x` are

Options

A. `1`
B. `10`
C. `10^(-5//2)`
D. `10^(6)`

Explanation

To find the possible values of \( x \) such that the third term in the expression \( (x + x^{\log_{10} x})^5 \) equals \( 10^6 \), we will follow these steps:

### Step 1: Identify the third term in the binomial expansion
The third term \( T_3 \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r-1} a^{n-(r-1)} b^{r-1} \]
For our expression, \( a = x \), \( b = x^{\log_{10} x} \), and \( n = 5 \). Thus, the third term \( T_3 \) is:
\[ T_3 = \binom{5}{2} x^{5-2} (x^{\log_{10} x})^2 \]

### Step 2: Calculate the coefficients and simplify
Calculating the coefficient:
\[ \binom{5}{2} = 10 \]
So, we have:
\[ T_3 = 10 \cdot x^3 \cdot (x^{\log_{10} x})^2 = 10 \cdot x^3 \cdot x^{2 \log_{10} x} = 10 \cdot x^{3 + 2 \log_{10} x} \]

### Step 3: Set the third term equal to \( 10^6 \)
We set the expression equal to \( 10^6 \):
\[ 10 \cdot x^{3 + 2 \log_{10} x} = 10^6 \]
Dividing both sides by 10 gives:
\[ x^{3 + 2 \log_{10} x} = 10^5 \]

### Step 4: Take logarithm on both sides
Taking logarithm base 10 on both sides:
\[ \log_{10}(x^{3 + 2 \log_{10} x}) = \log_{10}(10^5) \]
This simplifies to:
\[ (3 + 2 \log_{10} x) \cdot \log_{10} x = 5 \]

### Step 5: Substitute \( y = \log_{10} x \)
Let \( y = \log_{10} x \). Then the equation becomes:
\[ (3 + 2y) y = 5 \]
Expanding this gives:
\[ 2y^2 + 3y - 5 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = 3, c = -5 \):
\[ y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} \]
\[ y = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4} \]
Calculating the two possible values:
1. \( y = \frac{4}{4} = 1 \)
2. \( y = \frac{-10}{4} = -2.5 \)

### Step 7: Convert back to \( x \)
Recall \( y = \log_{10} x \):
1. For \( y = 1 \):
\[ \log_{10} x = 1 \implies x = 10 \]
2. For \( y = -2.5 \):
\[ \log_{10} x = -2.5 \implies x = 10^{-2.5} = \frac{1}{10^{2.5}} = 10^{-5/2} \]

### Final Answer
The possible values of \( x \) are:
\[ x = 10 \quad \text{and} \quad x = 10^{-5/2} \]