Using differentials, find the approximate value of `root(3)(0.026)`, upto three places of decimals.

Answer

To approximate the value of \(\sqrt[3]{0.026}\) using differentials, we can follow these steps:

### Step 1: Identify a nearby perfect cube
We know that \(0.027\) is a perfect cube since \(0.3^3 = 0.027\). Thus, we can express \(0.026\) as:
\[ 0.026 = 0.027 - 0.001 \]

### Step 2: Define the function and its derivative
Let \(y = x^{1/3}\), where \(x\) is the value we are interested in. The derivative of \(y\) with respect to \(x\) is:
\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} \]

### Step 3: Calculate the derivative at the point \(x = 0.027\)
Now, we evaluate the derivative at \(x = 0.027\):
\[ \frac{dy}{dx} \bigg|_{x=0.027} = \frac{1}{3} (0.027)^{-2/3} \]
Calculating \(0.027^{-2/3}\):
\[ 0.027^{1/3} = 0.3 \quad \text{and thus} \quad 0.027^{-2/3} = \frac{1}{0.3^2} = \frac{1}{0.09} = \frac{100}{9} \]
So,
\[ \frac{dy}{dx} \bigg|_{x=0.027} = \frac{1}{3} \cdot \frac{100}{9} = \frac{100}{27} \]

### Step 4: Calculate \(dy\) using \(\Delta x\)
We have \(\Delta x = -0.001\) (since we are moving from \(0.027\) to \(0.026\)). Thus,
\[ dy = \frac{dy}{dx} \cdot \Delta x = \frac{100}{27} \cdot (-0.001) = -\frac{100}{27000} = -\frac{1}{270} \]

### Step 5: Calculate the approximate value of \(\sqrt[3]{0.026}\)
Now, we can find the approximate value of \(\sqrt[3]{0.026}\):
\[ \sqrt[3]{0.026} \approx \sqrt[3]{0.027} + dy = 0.3 - \frac{1}{270} \]
Calculating \(\frac{1}{270}\):
\[ \frac{1}{270} \approx 0.0037 \]
Thus,
\[ \sqrt[3]{0.026} \approx 0.3 - 0.0037 = 0.2963 \]

### Step 6: Round to three decimal places
Finally, rounding \(0.2963\) to three decimal places gives:
\[ \sqrt[3]{0.026} \approx 0.296 \]

### Final Answer
The approximate value of \(\sqrt[3]{0.026}\) up to three decimal places is:
\[ \boxed{0.296} \]