Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(4 a^{2} b\right)^{4}\left(\frac{-a^{3}}{2 b}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( 64a^{14}b^2 \).

Step 1: Apply the Power of a Power Rule

For the expression \( (4a^2b)^4 \), use the power of a power rule: \( (x^m)^n = x^{mn} \). Applying this rule: \( 4^4 (a^2)^4 (b)^4 = 256a^8b^4 \). This simplifies the first part of the expression.

Step 2: Simplify the Second Expression

For the expression \( \left( \frac{-a^3}{2b} \right)^2 \), apply the power of a power rule to both the numerator and the denominator separately: \( (-a^3)^2 = (-1)^2(a^3)^2 = a^6 \) and \( (2b)^2 = 4b^2 \). This gives us \( \frac{a^6}{4b^2} \).

Step 3: Combine the Expressions

Multiply the results from Step 1 and Step 2: \( 256a^8b^4 \cdot \frac{a^6}{4b^2} \). Simplify by multiplying numerators and denominators separately: \( \frac{256a^8b^4a^6}{4b^2} = \frac{256a^{14}b^4}{4b^2} \).

Step 4: Simplify the Final Expression

Divide both terms by 4 and apply the rule: \( x^m / x^n = x^{m-n} \). Simplifying, you get \( 64a^{14}b^{2}\). The powers of \(b\) satisfy \( b^{4-2} = b^2 \). Hence, the final simplified expression is \( 64a^{14}b^2 \).

Key Concepts

Power of a Power Rule

When dealing with expressions that include exponents raised to another power, the "power of a power rule" is a powerful tool to simplify them efficiently. The rule states that for any base \(x\) raised to an exponent \(m\), which is then raised to another exponent \(n\), you multiply the exponents together: \[(x^m)^n = x^{mn}\] This means you take the base \(x\) and raise it to the power of the product of \(m\) and \(n\). In our exercise, we applied this rule to simplify each part of the larger expression. **Example:** - Consider the expression \((a^2)^4\). Using the rule, we simplify it to \(a^{8}\) because \(2 \times 4 = 8\). - Similarly, for the fraction \(\left(\frac{-a^3}{2b}\right)^2\), each part inside the parentheses is raised to the second power using the rule individually. Understanding and applying this rule correctly is essential for simplifying more complex expressions efficiently. Always remember to apply the exponent to each element within the parentheses.

Simplification

Simplification is the process of making an algebraic expression as simple as possible. This often involves reducing expressions to their most natural form. Here, the goal is to make the expression more manageable and easier to understand or evaluate. **Key Strategies for Simplification:** - **Combine like terms:** Same base and exponent terms are combined by adding or subtracting their coefficients. - **Use rules of exponents:** Such as the power of a power rule, product rule, and quotient rule to simplify powers. - **Cancel common factors:** Especially useful in fractions to reduce numerator and denominator. - **Rearrange terms:** To group and simplify mutually related terms. In our initial exercise, we simplified both parts of the expression separately before combining them again. This staged approach can help prevent errors and ensure each step is validated before merging into a final result.

Multiplication of Expressions

The multiplication of algebraic expressions often requires combining coefficients and applying exponent rules to like bases. It's an essential skill that needs practice, especially when dealing with expressions that involve fractions, negative signs, or multiple variables. **Steps for Successful Multiplication:**

• **Multiply Coefficients:** Simply multiply the numbers in the expression. For example, \(4 \times 256\).
• **Add Exponents of Like Bases:** When multiplying terms with the same base, add the exponents. For instance, \(a^8 \cdot a^6 = a^{8+6} = a^{14}\).
• **Simplify Resulting Expressions:** Once everything is multiplied, verify if further simplification is possible. Check for further factors to simplify fractions or powers to reduce to a more compact form.

In the final step of the provided solution, after multiplying together the simplified expressions from previous steps, we ensured everything was in its simplest form - a clear product without unnecessary complexity. Remember, practicing these steps will enhance your algebraic manipulation skills, making it easier to tackle more challenging problems.