Problem 60 - 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. $$ \left(-2 x y^{3}\right)^{-4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{16x^4y^{12}} \).

Step 1: Apply the Power Rule

The power rule for exponents states that \( (a^m)^n = a^{m\cdot n} \). Apply this to each factor in \( (-2xy^3)^{-4} \). The expression becomes \( (-2)^{-4}(x)^{-4}(y^{3})^{-4} \).

Step 2: Simplify Each Factor

Simplify each factor with the power rule. For \( (-2)^{-4} \), apply the exponent: \( -2^{-4} = (-1)^{-4}\cdot 2^{-4} = 1 \cdot 2^{-4} = \frac{1}{2^4} = \frac{1}{16} \). For \( (x)^{-4} \), it simplifies to \( \frac{1}{x^4} \). For \( (y^3)^{-4} \), use the power rule, \( (y^3)^{-4} = y^{-12} = \frac{1}{y^{12}} \).

Step 3: Combine the Simplified Factors

Combine the simplified expressions: \( \frac{1}{16} \cdot \frac{1}{x^4} \cdot \frac{1}{y^{12}} \). Multiply the denominators together: \[ \frac{1}{16x^4y^{12}}. \]

Step 4: Express the Final Answer

Rewrite the expression with positive exponents only: \[ \frac{1}{16x^4y^{12}}. \] This is already in simplest form with positive exponents.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to an exponent. This operation is fundamental in algebra and simplifies repeated multiplication. When you see an expression like \((-2xy^3)^{-4}\), it involves raising the whole expression to the power of \(-4\). Here, each component inside the parentheses, \(-2\), \(x\), and \(y^3\), must be raised individually to the power of \(-4\). Understanding exponentiation allows us to manipulate and simplify complex expressions easily. Remember that exponentiation follows specific rules, such as multiplying exponents when raising a power to a power, as seen here:

• Using the power rule: \((a^m)^n = a^{m \cdot n}\).
• Applying individually to each factor in the expression.

Exponentiation helps break down large expressions into manageable forms, aiding in deeper understanding and manipulation of algebraic problems. This simplification process is crucial before tackling more advanced algebraic operations.

Algebra Simplification

Algebra simplification involves using rules and operations to find the simplest form of an expression. This process typically reduces complexity, making expressions easier to work with. In our example, \((-2xy^3)^{-4}\), simplification requires us to handle each factor within the parenthesis and apply the exponent to each part. For instance:

• For \((-2)^{-4}\), compute as \((-1)^{-4} \cdot 2^{-4}\), which simplifies to \(\frac{1}{16}\).
• For \((x)^{-4}\), this becomes \(\frac{1}{x^4}\).
• For \((y^3)^{-4}\), it becomes \(y^{-12}\), which simplifies to \(\frac{1}{y^{12}}\).

These steps show how each part is simplified individually before combining them to form the complete expression. Simplification is vital as it leads to cleaner, more understandable solutions and makes further mathematical operations more manageable. It's an integral part of algebra, allowing for clarity and precision in problem-solving.

Negative Exponents

Negative exponents may seem challenging initially, but they have a simple interpretation. A negative exponent indicates that the base should be taken as a reciprocal, or inverted, and then raised to the positive of that exponent. This is helpful in converting expressions to a form that uses only positive exponents, as instructed in many problems. Consider the transformation in our exercise:

• \((-2)^{-4}\) becomes \(\frac{1}{16}\).
• \(x^{-4}\) is converted to \(\frac{1}{x^4}\).
• \(y^{-12}\) derived from \((y^3)^{-4}\), becomes \(\frac{1}{y^{12}}\).

Understanding how to convert negative exponents into fractions helps in rewriting expressions with positive exponents, simplifying further calculation or solving. Therefore, grasping the concept of negative exponents is crucial in making complex algebraic expressions workable and doing away with the negatives effectively.