Problem 17 - The Calculus with Analytic Geometry

Question

\(\sqrt[3]{0.00098}\)

Step-by-Step Solution

Answer

Approximately 0.097.

Step 1: Understand the Cube Root

The problem requires finding the cube root of a decimal number. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Step 2: Convert Decimal to Scientific Notation

First, convert the decimal number to scientific notation. Here, 0.00098 can be written as \( 9.8 \times 10^{-4} \).

Step 3: Apply the Cube Root to Scientific Notation

To find \( \sqrt[3]{9.8 \times 10^{-4}} \), apply the cube root to each part separately: \( \sqrt[3]{9.8} \) and \( \sqrt[3]{10^{-4}} \).

Step 4: Calculate Cube Root of the Decimal

Find \( \sqrt[3]{9.8} \) using a calculator, which is approximately 2.1.

Step 5: Calculate Cube Root of the Exponent

For the exponent part, use the property \( \sqrt[3]{10^{-4}} = 10^{-4/3} \), which simplifies to approximately \( 10^{-1.33} \).

Step 6: Combine the Results

Now, combine the results from Steps 4 and 5: \( 2.1 \times 10^{-1.33} \).

Step 7: Convert Back to Decimal Form

Convert the answer back to decimal form for the final result. \( 2.1 \times 10^{-1.33} \) is approximately 0.097.

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers, making them easier to work with. In scientific notation, a number is written as a product of two factors:

• a number greater than or equal to 1 and less than 10, and
• a power of 10.

For example, the number 0.00098 can be written in scientific notation as 9.8 × 10^{-4}. This format is particularly useful for simplifying calculations with very large or small numbers.

Decimal to Scientific Notation

Converting a decimal to scientific notation involves a few simple steps:

• Move the decimal point in the number until you have a number between 1 and 10. Count how many places you moved the decimal point.
• If you moved the decimal to the right, the exponent is negative. If you moved it to the left, the exponent is positive.

For example, to convert 0.00098 to scientific notation:

• Move the decimal point 4 places to the right to get 9.8.
• Since we moved it to the right, the exponent is -4.
• Therefore, 0.00098 in scientific notation is 9.8 × 10^{-4}.

Exponent Properties

Understanding exponent properties is crucial for working with powers and roots. Some key properties include:

• Product of Powers: a^m × a^n = a^{m+n}.
• Power of a Power: (a^m)^n = a^{m×n}.
• Power of a Product: (ab)^n = a^n × b^n.
• Roots and Radicals: n√(a^m) = a^{m/n).

In our exercise, we specifically used the property for roots: √(10^{-4}) = 10^{−4/n}. This helps in simplifying complex expressions involving roots and exponents.

Cube Root of a Decimal

Finding the cube root of a decimal can seem tricky, but breaking it down step by step helps. Let's continue with our solved problem √(0.00098):
1. First, convert the decimal to scientific notation: 0.00098 = 9.8 × 10^{-4}.
2. Apply the cube root to each part separately: ( √(9.8) and √10^{-4}).

• √(9.8) is approximately 2.1.
• √10^{-4} simplifies to 10^{-4/3} = 10^{-1.33}.

Combining these results, we get approximately 2.1 × 10^{-1.33}, which converts to 0.097 in decimal form. This step-by-step approach simplifies the process of finding the cube root of decimals.