Evaluating Cube Root of Square Root

To solve the expression \(\sqrt[3]{{\sqrt {0.000729} }}\), we need to perform the operations from the inside out. This means we will first calculate the square root of 0.000729 and then find the cube root of that result.

Square Root Calculation

First, let us calculate the square root of \(0.000729\).

• We can write the decimal number as a fraction: \(0.000729 = \frac{729}{1000000}\).
• Now, we need to find the square root of the numerator and the denominator separately.
• We know that \(27 \times 27 = 729\), so \(\sqrt{729} = 27\).
• We also know that \(1000 \times 1000 = 1000000\), so \(\sqrt{1000000} = 1000\).
• Therefore, \(\sqrt{0.000729} = \sqrt{\frac{729}{1000000}} = \frac{\sqrt{729}}{\sqrt{1000000}} = \frac{27}{1000}\).
• Converting this fraction back to a decimal, we get \(\frac{27}{1000} = 0.027\).

So, the expression simplifies to \(\sqrt[3]{{0.027}}\).

Cube Root Calculation

Next, we need to calculate the cube root of \(0.027\).

• Again, we can write the decimal number as a fraction: \(0.027 = \frac{27}{1000}\).
• Now, we need to find the cube root of the numerator and the denominator separately.
• We know that \(3 \times 3 \times 3 = 27\), so \(\sqrt[3]{27} = 3\).
• We also know that \(10 \times 10 \times 10 = 1000\), so \(\sqrt[3]{1000} = 10\).
• Therefore, \(\sqrt[3]{0.027} = \sqrt[3]{\frac{27}{1000}} = \frac{\sqrt[3]{27}}{\sqrt[3]{1000}} = \frac{3}{10}\).
• Converting this fraction back to a decimal, we get \(\frac{3}{10} = 0.3\).

Thus, the final value of the expression \(\sqrt[3]{{\sqrt {0.000729} }}\) is \(0.3\).

Final Answer Verification

Let's summarize the steps:

1. Start with the expression: \(\sqrt[3]{{\sqrt {0.000729} }}\)
2. Calculate the inner square root: \(\sqrt{0.000729} = 0.027\)
3. Calculate the outer cube root: \(\sqrt[3]{0.027} = 0.3\)

The result, \(0.3\), matches one of the given options.