Cube Root of 0.027 Explained

Finding the cube root of a decimal number like 0.027 involves understanding how to convert decimals into fractions and then applying the cube root property to both the numerator and the denominator. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Understanding Cube Roots

• A cube root is the inverse operation of cubing a number. For example, since \(3^3 = 3 \times 3 \times 3 = 27\), the cube root of 27 is 3.
• The symbol for a cube root is \(\sqrt[3]{\phantom{x}}\).

Step-by-Step Calculation of the Cube Root of 0.027

To find the cube root of 0.027, we can follow these steps:

1. Convert the Decimal to a Fraction:

   The decimal 0.027 can be written as a fraction. Since there are three digits after the decimal point, we can write it over 1000:

   $$0.027 = \frac{27}{1000}$$

2. Apply the Cube Root Property:

   Now, we need to find the cube root of this fraction. The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator:

   $$\sqrt[3]{0.027} = \sqrt[3]{\frac{27}{1000}} = \frac{\sqrt[3]{27}}{\sqrt[3]{1000}}$$

3. Find the Cube Root of the Numerator:

   We need to find a number that, when multiplied by itself three times, equals 27.

   • \(1 \times 1 \times 1 = 1\)
   • \(2 \times 2 \times 2 = 8\)
   • \(3 \times 3 \times 3 = 27\)

   So, the cube root of 27 is 3:

   $$\sqrt[3]{27} = 3$$

4. Find the Cube Root of the Denominator:

   Next, we find a number that, when multiplied by itself three times, equals 1000.

   • \(10 \times 10 \times 10 = 1000\)

   So, the cube root of 1000 is 10:

   $$\sqrt[3]{1000} = 10$$

5. Combine the Results:

   Now, substitute the cube roots back into the fraction:

   $$\frac{\sqrt[3]{27}}{\sqrt[3]{1000}} = \frac{3}{10}$$

6. Convert the Fraction Back to a Decimal:

   Finally, convert the fraction \(\frac{3}{10}\) back to its decimal form:

   $$\frac{3}{10} = 0.3$$

Final Result for 0.027 Cube Root

Therefore, the cube root of 0.027 is 0.3.

To verify, we can multiply 0.3 by itself three times:

$$0.3 \times 0.3 \times 0.3 = 0.09 \times 0.3 = 0.027$$

This confirms our calculation.