Problem 28 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt[3]{24 w^{2}}}{\sqrt[3]{3 w^{4}}} $$

Step-by-Step Solution

Answer

The quotient simplifies to \( \frac{2}{w^{2/3}} \).

Step 1: Simplify the Radicals

First, let's express the radicals in both the numerator and denominator as powers. We have \( \sqrt[3]{24w^2} = (24w^2)^{1/3} \) and \( \sqrt[3]{3w^4} = (3w^4)^{1/3} \). The quotient becomes \( \frac{(24w^2)^{1/3}}{(3w^4)^{1/3}} \).

Step 2: Apply the Quotient Rule for Exponents

Using the property that \( \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \), we can rewrite the expression as \( \left( \frac{24w^2}{3w^4} \right)^{1/3} \). Simplify \( \frac{24w^2}{3w^4} \) inside the parentheses.

Step 3: Simplify the Fraction Inside the Parentheses

Simplify \( \frac{24w^2}{3w^4} \) by dividing the coefficients and subtracting exponents for the variables. The fraction inside becomes \( \frac{24}{3} \cdot \frac{w^2}{w^4} = 8w^{-2} \).

Step 4: Apply the Cube Root to the Simplified Expression

Now take the cube root of \( 8w^{-2} \): \( (8w^{-2})^{1/3} = 8^{1/3} \cdot w^{-2/3} \). Since \( 8^{1/3} = 2 \), the expression simplifies to \( 2w^{-2/3} \).

Step 5: Finalize the Simplified Form

Convert \( 2w^{-2/3} \) into a more conventional form by writing the negative exponent as a fraction: \( \frac{2}{w^{2/3}} \). This is the simplified quotient.

Key Concepts

Understanding Cube Roots

Cube roots are the inverse operation of cubing a number. When you cube a number, you multiply it by itself three times. A cube root asks the question: "What number, when multiplied by itself three times, gives us this original number?"

For example, the cube root of 8 is 2 because

• 2 multiplied by itself three times (2 × 2 × 2) equals 8.

Cube roots are often represented with a radical sign and an index. The index of 3 tells us we are looking for a cube root.

• For example, \(\sqrt[3]{27} = 3\) because \(3 imes 3 imes 3 = 27\).
• In algebraic expressions, cube roots simplify terms just like they do with numbers.

Mastering the Quotient Rule for Exponents

The quotient rule for exponents is a helpful mathematical property when dealing with division of powers. It states that if we have the same base, we can subtract the exponents.

In mathematical terms, \(\frac{a^m}{a^n} = a^{m - n}\). This is very useful when simplifying expressions with exponents.

For example, consider \(\frac{w^5}{w^2}\). Applying the rule, we subtract the exponents such that it becomes \(w^{5-2} = w^3\).

When the bases are the same, remember these steps:

• Subtract the bottom exponent from the top exponent.
• Keep the base the same in the result.
• If the result is zero, the outcome is 1.

Dealing with Negative Exponents

Negative exponents can be tricky at first, but they actually have a simple meaning. A negative exponent indicates that the base should be taken as the reciprocal and then apply the positive exponent.

For example, \(2^{-3}\) is equivalent to \(\frac{1}{2^3}\). Therefore, \(2^{-3} = \frac{1}{8}\).

Whenever you encounter a negative exponent, follow these steps:

• Turn it into a positive exponent by flipping it to the denominator (or numerator if it's already in denominator).
• Calculate the positive exponent normally.
• Place the result in the correct position (either as a fraction or part of a fraction).

Simplifying Radicals

Simplifying radicals involves reducing the expression to its most basic form while still preserving its value.

To simplify a radical expression:

• Factor the number or expression inside the radical as much as possible.
• Look for perfect squares (or perfect cubes depending on the root) among these factors.
• Rewrite the expression by taking these perfect powers out of the radical.

For example, in the expression \(\sqrt[3]{8w^2}\), we recognize that 8 is a perfect cube, since \(2^3 = 8\). So, it simplifies to \(2w^{2/3}\).

The key is recognizing perfect powers and simplifying each part of the expression systematically. Practice helps in getting faster and more accurate!