The expansion by the binomial theorem of `(2 x + (1)/(8) )^(10)` is `1024x^(10) + 640x^(9) + ax^(8) + bx^(7) + `... Calculate
(i) the numerical value of a and b

Answer

To solve the problem, we will use the Binomial Theorem to find the coefficients \(a\) and \(b\) in the expansion of \((2x + \frac{1}{8})^{10}\).

### Step 1: Write the Binomial Expansion
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 2x\), \(b = \frac{1}{8}\), and \(n = 10\). Thus, the expansion is:
\[ (2x + \frac{1}{8})^{10} = \sum_{r=0}^{10} \binom{10}{r} (2x)^{10-r} \left(\frac{1}{8}\right)^r \]

### Step 2: Find the Coefficient \(a\) for \(x^8\)
To find the coefficient of \(x^8\), we need \(10 - r = 8\) which gives \(r = 2\). The term corresponding to \(r = 2\) is:
\[ T_{2+1} = \binom{10}{2} (2x)^{10-2} \left(\frac{1}{8}\right)^2 \]
Calculating this:
\[ = \binom{10}{2} (2x)^8 \left(\frac{1}{8}\right)^2 \]
\[ = \binom{10}{2} (2^8 x^8) \left(\frac{1}{64}\right) \]
\[ = \binom{10}{2} \cdot 256 x^8 \cdot \frac{1}{64} \]
\[ = \binom{10}{2} \cdot 4 x^8 \]
Now, calculating \(\binom{10}{2}\):
\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]
Thus,
\[ = 45 \cdot 4 x^8 = 180 x^8 \]
So, the coefficient \(a = 180\).

### Step 3: Find the Coefficient \(b\) for \(x^7\)
To find the coefficient of \(x^7\), we need \(10 - r = 7\) which gives \(r = 3\). The term corresponding to \(r = 3\) is:
\[ T_{3+1} = \binom{10}{3} (2x)^{10-3} \left(\frac{1}{8}\right)^3 \]
Calculating this:
\[ = \binom{10}{3} (2x)^7 \left(\frac{1}{8}\right)^3 \]
\[ = \binom{10}{3} (2^7 x^7) \left(\frac{1}{512}\right) \]
\[ = \binom{10}{3} \cdot 128 x^7 \cdot \frac{1}{512} \]
\[ = \binom{10}{3} \cdot \frac{128}{512} x^7 \]
\[ = \binom{10}{3} \cdot \frac{1}{4} x^7 \]
Now, calculating \(\binom{10}{3}\):
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Thus,
\[ = 120 \cdot \frac{1}{4} x^7 = 30 x^7 \]
So, the coefficient \(b = 30\).

### Final Answer
The numerical values are:
\[ a = 180, \quad b = 30 \]