Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[3]{8 a^{6} b^{-3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2a^2}{b} \).

Step 1: Understand the Expression

The expression given to simplify is \( \sqrt[3]{8 a^{6} b^{-3}} \), which lies within a cube root. Our goal is to simplify this cube root.

Step 2: Break Down the Expression into Groups

Recognize that \( 8 \) can be rewritten as \( 2^3 \). Thus, the expression inside the cube root \( \sqrt[3]{8 a^6 b^{-3}} \) can be rewritten as \( \sqrt[3]{2^3 a^6 b^{-3}} \).

Step 3: Apply Cube Root to Each Component

Applying the cube root to each part, we have: \( \sqrt[3]{2^3} \), \( \sqrt[3]{a^6} \), and \( \sqrt[3]{b^{-3}} \). Simplifying each gives: \( 2 \), \( a^2 \), and \( b^{-1} \).

Step 4: Combine the Results

Combine the simplified parts to form the solution for the original expression: \( 2a^2b^{-1} \), which can be written as \( \frac{2a^2}{b} \).

Key Concepts

Cube Root

The cube root symbol, written as \( \sqrt[3]{} \), is used to determine a number which, when multiplied by itself three times, gives the original number under the cube root. For example, if we have \( \sqrt[3]{8} \), we are seeking a number which multiplied by itself three times results in 8. In this case, 2 is our answer because \( 2 \times 2 \times 2 = 8 \).

Understanding the cube root is important when simplifying expressions. To break it down:

• First, identify if any numbers are perfect cubes (like 1, 8, 27, etc.).
• If they are, simplify them by taking their cube root.
• Apply this simplified logic to variables with exponents, such as turning \( a^6 \) into \( a^2 \) while using cube roots.

This way, you make expressions less complex, thereby achieving a simplified form.

Rationalize the Denominator

Rationalizing the denominator means adjusting the expression so that the denominator no longer contains radicals (such as square roots or cube roots). This is essential as it makes mathematical expressions easier to read and work with.

To rationalize:

• Multiply both the numerator and the denominator by a number that eliminates the radical in the denominator.
• The goal is to make the denominator a rational number.

For example, if you have a denominator like \( \sqrt[3]{b} \), you would want to eliminate the cube root by multiplying. In the example provided, transforming \( b^{-1} \) in the expression to \( \frac{1}{b} \) results in a rationalized format with no cube root in the denominator.

When you rationalize, the process involves balancing by using the inverse of any present exponents when necessary, ensuring expressions remain valid yet simpler.

Exponents

Exponents denote repeated multiplication of a base number. They are crucial for simplifying expressions and are represented by a small number placed to the top right of a base number, like \( a^6 \). This expression indicates that \( a \) is to be multiplied by itself six times.

Exponents are used comprehensively in algebra to make expressions succinct and efficient for calculations.

• Multiplying powers with the same base adds the exponents: \( a^m \times a^n = a^{m+n} \).
• Dividing powers with the same base subtracts the exponents: \( a^m \div a^n = a^{m-n} \).
• Negative exponents indicate division: \( a^{-n} = \frac{1}{a^n} \).

When simplifying using exponents, it's useful to recognize patterns of powers and roots. In the original solution, \( a^6 \) is reinterpreted through its cube root as \( a^2 \), illustrating how exponents simplify complex root expressions. Understanding these rules helps in transforming expressions into a more concise and manageable form.