Problem 51 - College Algebra

Question

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth. $$\left(\frac{12^{3} m^{33}}{4^{-3}}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \( 12,214,323,584 m^{66} \).

Step 1: Simplify the Fraction Inside the Parentheses

Simplify the fraction \( \frac{12^3 m^{33}}{4^{-3}} \). First, simplify the numerical part: \( \frac{12^3}{4^{-3}} \). Since \( 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \), the fraction becomes \( 12^3 \times 64 \). Calculate \( 12^3 = 1728 \), then multiply: \( 1728 \times 64 = 110592 \). So the expression inside becomes \( 110592 m^{33} \).

Step 2: Apply the Power of a Power Rule

Raise the expression \( (110592 m^{33}) \) to the power of 2. Use the power of a power rule \( (ab)^n = a^n b^n \). Thus, the expression becomes \((110592)^2 (m^{33})^2 = 110592^2 m^{66}\).

Step 3: Calculate Numerical Power

Calculate \( 110592^2 \) using a calculator or similar tool to simplify. This results in the number \( 12,214,323,584 \).

Step 4: Combine the Results

The expression simplifies to \( 12,214,323,584 m^{66} \). There are no further simplifications, so this is the final answer.

Key Concepts

Graphing Calculator

A graphing calculator can be a powerful tool when performing complex algebraic simplifications. It not only helps with visualizing functions but also makes the computation of large numbers much more manageable. In our exercise, the graphing calculator assists by computing the powers of large numbers such as \(110592^2\) quickly and accurately.

Usage tips for a graphing calculator include:

• Entering formulas exactly as they appear to avoid any computational errors.
• Using the calculator's ability to store and recall numbers to handle intermediate results efficiently.
• Understanding how your specific model handles exponentiation to ensure accuracy.

With these features, a graphing calculator turns a potentially complex simplification exercise into a straightforward task.

Power of a Power Rule

The power of a power rule is a fundamental concept in algebra. It states that when raising a power to another power, you can multiply the exponents: \((a^m)^n = a^{m \times n}\). This rule is handy when simplifying expressions involving exponents.

In the given exercise, after simplifying the initial fraction, we apply this rule to \((110592 m^{33})^2\), resulting in \(110592^2\) and \((m^{33})^2 = m^{66}\). Each term is raised to the corresponding power independently, dramatically simplifying the expression.

Remembering this rule will aid in swiftly handling more complex exponential expressions and helps to understand how different terms combine under exponentiation.

Numerical Computation

Numerical computation involves calculating exact numerical values from algebraic expressions. This often includes handling large numbers with ease. In our exercise, we used numerical computation to get the value of \(110592^2\) which calculated to \(12,214,323,584\).

Key aspects of effective numerical computation include:

• Using proper tools like graphing calculators, which can handle immense numerical values reliably.
• Ensuring all calculations maintain units or variables as needed, like keeping \(m^{66}\) attached in our solution.
• Being aware of rounding methods if an exact numerical answer isn't required, ensuring precision in cases where necessary.

Mastering numerical computation can solve even highly complex problems in a fraction of the time it might take by hand.

Exponent Rules

Exponent rules are a foundation of algebra, providing the guidelines for simplifying and manipulating expressions involving powers. These rules detail how exponents act when multiplied, divided, or raised to another power.

Important exponent rules include:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)
• Negative exponents: \(a^{-n} = \frac{1}{a^n}\)

In our exercise, while \(4^{-3}\) was simplified using negative exponents, the power of a power rule provided a seamless transition in further simplifications.

Remember to practice these rules frequently as they form the basis for more advanced topics in mathematics, helping to simplify complex expressions fluently.