Write down the expansion by the binomial theorem of `(3x - (y)/(2) )^(4)`. By giving `x and y` suitable values, deduce the value of `(29.5)^(4)` correct to four significant figures.

Answer

To solve the problem, we will follow the steps outlined in the video transcript and provide a detailed step-by-step solution.

### Step 1: Write down the binomial expansion of \((3x - \frac{y}{2})^4\)

Using the binomial theorem, the expansion of \((a + b)^n\) is given by:

\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

For our expression \((3x - \frac{y}{2})^4\), we can identify \(a = 3x\) and \(b = -\frac{y}{2}\), and \(n = 4\).

The expansion is:

\[ (3x - \frac{y}{2})^4 = \sum_{k=0}^{4} \binom{4}{k} (3x)^{4-k} \left(-\frac{y}{2}\right)^k \]

Calculating each term:

- For \(k=0\): \(\binom{4}{0} (3x)^4 \left(-\frac{y}{2}\right)^0 = 1 \cdot (3x)^4 = 81x^4\)
- For \(k=1\): \(\binom{4}{1} (3x)^3 \left(-\frac{y}{2}\right)^1 = 4 \cdot (27x^3) \cdot \left(-\frac{y}{2}\right) = -54x^3y\)
- For \(k=2\): \(\binom{4}{2} (3x)^2 \left(-\frac{y}{2}\right)^2 = 6 \cdot (9x^2) \cdot \frac{y^2}{4} = \frac{54}{4} x^2 y^2 = 13.5x^2y^2\)
- For \(k=3\): \(\binom{4}{3} (3x)^1 \left(-\frac{y}{2}\right)^3 = 4 \cdot (3x) \cdot \left(-\frac{y^3}{8}\right) = -\frac{12xy^3}{8} = -1.5xy^3\)
- For \(k=4\): \(\binom{4}{4} (3x)^0 \left(-\frac{y}{2}\right)^4 = 1 \cdot \frac{y^4}{16} = \frac{y^4}{16}\)

Putting it all together, we have:

\[ (3x - \frac{y}{2})^4 = 81x^4 - 54x^3y + 13.5x^2y^2 - 1.5xy^3 + \frac{y^4}{16} \]

### Step 2: Substitute suitable values for \(x\) and \(y\) to deduce \((29.5)^4\)

We can express \(29.5\) as \(30 - 0.5\). Thus, we set:

- \(3x = 30\) → \(x = 10\)
- \(-\frac{y}{2} = -0.5\) → \(y = 1\)

Now substituting \(x = 10\) and \(y = 1\) into the expansion:

\[ (3(10) - \frac{1}{2})^4 = 81(10)^4 - 54(10)^3(1) + 13.5(10)^2(1)^2 - 1.5(10)(1)^3 + \frac{(1)^4}{16} \]

Calculating each term:

1. \(81(10^4) = 810000\)
2. \(-54(10^3) = -54000\)
3. \(13.5(10^2) = 1350\)
4. \(-1.5(10) = -15\)
5. \(\frac{1}{16} = 0.0625\)

Now, summing these values:

\[ 810000 - 54000 + 1350 - 15 + 0.0625 \]

Calculating step-by-step:

- \(810000 - 54000 = 756000\)
- \(756000 + 1350 = 757350\)
- \(757350 - 15 = 757335\)
- \(757335 + 0.0625 = 757335.0625\)

### Step 3: Round to four significant figures

The value \(757335.0625\) rounded to four significant figures is \(757300\).

### Final Answer

Thus, the value of \((29.5)^4\) correct to four significant figures is:

\[ \boxed{757300} \]