If u+t=5 and u-t=2 , what is the value of `(u-t)(u^2-t^2)` ?

Explanation

To solve the problem, we need to find the value of \((u-t)(u^2-t^2)\) given the equations \(u + t = 5\) and \(u - t = 2\).

### Step-by-Step Solution:

1. **Identify the expressions:**
We know that \(u^2 - t^2\) can be factored using the difference of squares formula:
\[ u^2 - t^2 = (u + t)(u - t) \]
Therefore, we can rewrite our expression:
\[ (u - t)(u^2 - t^2) = (u - t)((u + t)(u - t)) \]

2. **Simplify the expression:**
This can be simplified to:
\[ (u - t)^2 (u + t) \]

3. **Substitute the known values:**
From the problem, we have:
- \(u + t = 5\)
- \(u - t = 2\)

Now substituting these values into the expression:
\[ (u - t)^2 = (2)^2 = 4 \]
Therefore:
\[ (u - t)^2 (u + t) = 4 \cdot 5 \]

4. **Calculate the final result:**
Now, we can calculate:
\[ 4 \cdot 5 = 20 \]

Thus, the value of \((u-t)(u^2-t^2)\) is **20**.