Problem 61 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{4}{x}\right)^{3} $$

Step-by-Step Solution

Answer

\( \frac{64}{x^3} \)

Step 1: Apply the Power Rule to the Fraction

We are given the expression \( \left(\frac{4}{x}\right)^3 \). The power rule for exponents states \( (a/b)^m = a^m / b^m \). Apply this rule to distribute the exponent of 3 to both the numerator and the denominator: \[ \left(\frac{4}{x}\right)^3 = \frac{4^3}{x^3}. \]

Step 2: Raise the Numerator to the Power

Next, we calculate \(4^3\). This means multiplying 4 by itself three times: \[ 4^3 = 4 \times 4 \times 4 = 64. \] Therefore, the numerator becomes 64.

Step 3: Simplify the Expression

Substitute the calculated values back into the fraction: \[ \frac{4^3}{x^3} = \frac{64}{x^3}. \] Now the expression is simplified using positive exponents.

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics that describe how many times a number, known as the base, is multiplied by itself. Think of an exponent as a shorthand for repeated multiplication. For example, in the expression \(4^3\), the number 4 is the base, and the exponent 3 tells us to multiply 4 by itself three times: \(4 \times 4 \times 4 = 64\).
Exponents are important in simplifying expressions and making calculations more manageable. They follow specific rules that help in arithmetic operations, such as:

• Power rule: \((a^m)^n = a^{m \times n}\). This means raising a power to another power involves multiplying the exponents.
• Product rule: \(a^m \times a^n = a^{m+n}\). This states that when multiplying like bases, you add the exponents.
• Zero exponent rule: \(a^0 = 1\). Any non-zero base raised to the power of zero is equal to 1.

Understanding these rules is essential as they are often used in simplifying complex expressions, especially when fractions and variables are involved.

Fractions

Fractions represent a part of a whole and consist of two parts: a numerator and a denominator. The numerator is the top part of the fraction, indicating how many parts we have, and the denominator is the bottom part, indicating the total number of equal parts in a whole.
When working with fractions in expressions, like \(\left(\frac{4}{x}\right)^3\), it is important to apply the exponent to both the numerator and the denominator. This is where understanding the power rule for fractions becomes essential.

• Applying powers to fractions: The rule \((a/b)^m = a^m/b^m\) helps distribute the exponent to both the numerator and the denominator, simplifying the expression efficiently.
• Simplification: Simplifying fractions before or after applying the power can make calculations easier. For example, reducing a fraction before raising it to a power can simplify the final results.

By mastering the manipulation of exponents and fractions, students can tackle a wide range of problems involving these two concepts.

Simplification

Simplification is the process of reducing an expression to its simplest form, making it easier to understand and work with. When dealing with exponents and fractions, simplification often follows a step-by-step approach to ensure accuracy and clarity.
In the expression \(\frac{64}{x^3}\), simplification involves:

• Ensure all exponents are positive: Expressions should be rewritten using positive exponents. This makes understanding the value and meaning of terms straightforward.
• Combine like terms: In cases where multiple similar terms exist, try to combine them to reduce the overall complexity of the expression.
• Reduce fractions: If possible, simplify the fraction further by dividing both the numerator and the denominator by their greatest common factor.

Through careful simplification, mathematical expressions become more manageable, paving the way for smoother problem-solving and deeper understanding of complex concepts.