Calculate ,if n=-(1)/(2) (5n)^3:25n^6=

To solve the expression \((5n)^3 : 25n^6\) when \(n = -\frac{1}{2}\), we need to follow these steps:<br /><br />1. Substitute \(n = -\frac{1}{2}\) into the expression.<br />2. Simplify the numerator and the denominator separately.<br />3. Divide the simplified numerator by the simplified denominator.<br /><br />Let's go through these steps in detail:<br /><br />### Step 1: Substitute \(n = -\frac{1}{2}\)<br /><br />\[<br />(5n)^3 : 25n^6 \quad \text{becomes} \quad \left(5 \left(-\frac{1}{2}\right)\right)^3 : 25 \left(-\frac{1}{2}\right)^6<br />\]<br /><br />### Step 2: Simplify the numerator and the denominator<br /><br />#### Numerator:<br />\[<br />(5 \cdot -\frac{1}{2})^3 = \left(-\frac{5}{2}\right)^3 = -\frac{125}{8}<br />\]<br /><br />#### Denominator:<br />\[<br />25 \left(-\frac{1}{2}\right)^6 = 25 \left(\frac{1}{64}\right) = \frac{25}{64}<br />\]<br /><br />### Step 3: Divide the simplified numerator by the simplified denominator<br /><br />\[<br />\left(-\frac{125}{8}\right) : \left(\frac{25}{64}\right) = \left(-\frac{125}{8}\right) \times \left(\frac{64}{25}\right) = -\frac{125 \times 64}{8 \times 25} = -\frac{8000}{200} = -40<br />\]<br /><br />Therefore, the value of the expression \((5n)^3 : 25n^6\) when \(n = -\frac{1}{2}\) is \(-40\).