Problem 47 - Algebra for College Students

Question

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt[3]{8000}\)

Step-by-Step Solution

Answer

\(\sqrt[3]{8000} = 20\).

Step 1: Express 8000 in Scientific Notation

First, express 8000 as a power of 10. The number 8000 can be written as \(8 \times 10^3\). This is the scientific notation for 8000 because \(8 \times 10^3 = 8000\).

Step 2: Apply the Cube Root to Scientific Notation

Now we need to apply the cube root: \(\sqrt[3]{8 \times 10^3}\). According to properties of exponents, this is equivalent to \(\sqrt[3]{8} \times \sqrt[3]{10^3}\).

Step 3: Simplify the Cube Root of 8

\(\sqrt[3]{8}\) is the cube root of 8. Since \(2^3 = 8\), it follows that \(\sqrt[3]{8} = 2\).

Step 4: Simplify the Cube Root of \(10^3\)

Find the cube root of \(10^3\). This simplifies to \((10^3)^{1/3} = 10^{3/3} = 10^1 = 10\).

Step 5: Multiply the Results Together

Now multiply the results from Steps 3 and 4. \(2 \times 10 = 20\). Thus, \(\sqrt[3]{8000} = 20\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with scientific notation, especially for simplifying expressions like cube roots. Exponents allow us to express numbers in a compact form which makes calculations easier to manage. When you have an expression such as \((a \times b)^n\), it can be simplified using the rule \(a^n \times b^n\). This is precisely what happens in the example of finding the cube root of 8000 expressed in scientific notation.

Here are some essential properties to remember:

• Product rule: \(a^m \times a^n = a^{m+n}\) - This combines exponents with the same base.
• Power of a power rule: \((a^m)^n = a^{m \times n}\) - This is used to simplify an exponent raised to another exponent.
• Quotient rule: \(\left(\frac{a^m}{a^n}\right) = a^{m-n}\) - This simplifies division expressions with the same base.
• Cubic root as an exponent: \((a^m)^{1/3} = a^{m/3}\) - This is specifically useful for cube roots.

These properties allow us to switch effortlessly between different expressions, making calculations including roots and powers much more efficient.

Cube Root

The cube root is the number that, when multiplied by itself twice, gives the original number. For instance, the cube root of 8 is 2 because \(2^3 = 8\). In scientific notation, cube roots can be simplified using exponent rules.

For calculating cube roots of numbers in scientific notation like \(8 \times 10^3\):

• First, you take the cube root of the coefficient, in this case, 8, which is 2 since \(2^3 = 8\).
• Then, calculate the cube root of the power of 10. The cube root of \(10^3\) is \(10^{3/3} = 10^1 = 10\).
• Finally, multiply these results to get the answer. Here, \(2 \times 10 = 20\).

The process exemplifies how understanding roots like the cubic root are useful in breaking down otherwise complex expressions into simple calculations.

Simplifying Expressions

Simplifying expressions is a valuable skill, particularly when dealing with large numbers or scientific notation. The goal is to reduce expressions to their simplest form, making them easier to work with and understand. Simplifying the cube root of a number like 8000 requires breaking it down into understandable parts.

Here's how you simplify expressions using the properties of exponents and cube roots:

• Express in simpler terms: Convert the number to scientific notation, which in this case means expressing 8000 as \(8 \times 10^3\). This step sets the stage for easy manipulation with roots.
• Break into components: When simplifying \(\sqrt[3]{8 \times 10^3}\), use exponent rules to separate the coefficient from the power of ten. This gives us \(\sqrt[3]{8} \times \sqrt[3]{10^3}\).
• Simplify each part individually: Calculate the cube root of each part separately. \(\sqrt[3]{8}\) simplifies to 2, and \(\sqrt[3]{10^3}\) simplifies to 10.
• Combine and conclude: Multiply these simplified results to obtain the final simplified expression. Here, \(2 \times 10 = 20\), so \(\sqrt[3]{8000} = 20\).

This practical approach aids in understanding and applying mathematical concepts effectively, ensuring even complex expressions are manageable.