Introduction:

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, we need to find the cube root of 0.8.

Method:

To find the cube root of 0.8, we can use various methods such as prime factorization, estimation, or using a calculator. Let's use the prime factorization method.

Prime Factorization of 0.8:

To begin, let's express 0.8 as a fraction. Since 0.8 can be written as 8/10, we can simplify it further by dividing both the numerator and denominator by their greatest common divisor, which is 2. This gives us 4/5.

Using Prime Factorization:

Now, let's find the prime factors of both the numerator and the denominator.

Prime factors of 4: 2 * 2
Prime factors of 5: 5

Simplifying:

We can cancel out the common prime factor of 2 from the numerator and denominator, resulting in 2/5.

The Cube Root of 0.8:

Now that we have simplified the fraction to 2/5, we can find its cube root.

The cube root of 2/5 is ∛(2/5).

Calculating the Cube Root:

To calculate the cube root, you can use a calculator or estimation techniques. Using a calculator, the cube root of 2/5 is approximately 0.62996.

Therefore, the cube root of 0.8 is approximately 0.62996.

Summary:

To summarize the steps:
1. Express 0.8 as a fraction, which gives us 4/5.
2. Find the prime factors of both the numerator and the denominator.
3. Cancel out the common prime factors to simplify the fraction to 2/5.
4. Calculate the cube root of 2/5 using a calculator or estimation techniques.
5. The cube root of 0.8 is approximately 0.62996.

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