Problem 51 - College Algebra with Corequisite Support

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 \(12230590464 m^{66}\).

Step 1: Simplify the Inner Fraction

First, simplify the expression inside the parentheses: \[ \frac{12^3 m^{33}}{4^{-3}} \]This expression is equivalent to multiplying the numerator and denominator, i.e., \[ 12^3 m^{33} \times 4^3 \]since we can change \(\frac{1}{4^{-3}}\) to \(4^3\).

Step 2: Calculate the Powers

Calculate the powers in the expression:- \(12^3 = 12 \times 12 \times 12 = 1728\)- \(4^3 = 4 \times 4 \times 4 = 64\)Now substitute these values back into the expression:\[ 1728 \times 64 \times m^{33} \]

Step 3: Multiply the Results

Next, multiply \(1728\) by \(64\):\[ 1728 \times 64 = 110592 \]Thus, the expression becomes \[ 110592 m^{33} \].

Step 4: Square the Entire Expression

Now, square the entire expression:\[(110592 m^{33})^2 = 110592^2 \times (m^{33})^2 \]- \((m^{33})^2 = m^{66}\)- Perform \(110592^2\) using a calculator to get the numerical value for the constant: \(110592^2 = 12230590464\) Hence, the expression becomes: \[ 12230590464 m^{66} \]

Step 5: Prepare the Final Answer in Decimal Form

Since the question asks for rounding to the nearest hundredth, we'll convert the number into decimal form using a calculator if necessary. However, since this is already a whole number expression without decimals in this format, consider the power of m:\[12230590464 m^{66}\].

Key Concepts

Power Manipulation

Power manipulation involves handling expressions with exponents in such a way that it simplifies the calculation. When given an expression like \( \left(\frac{12^{3} m^{33}}{4^{-3}}\right)^{2} \), the goal is to simplify the complexity by addressing the powers involved.

This generally involves recognizing full powers and fractions, such as transforming \( \frac{1}{4^{-3}} \) into \( 4^3 \). When manipulating powers, you can:

• Recognize inversion properties such as \( a^{-n} = \frac{1}{a^n} \)
• Combine powers by multiplying bases and adding exponents
• Convert multiplication to simple calculations using power identities
• Simplify inside the parentheses before other operations

By understanding these aspects, you can reduce the time and effort needed to solve complex algebraic problems.

Exponent Rules

Exponent rules are key when dealing with expressions that involve powers. They help us understand how to handle and operate on terms that include exponents. Here are some important rules:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)
• Zero Exponent Rule: \( a^0 = 1 \) if \( a eq 0 \)

In the context of our problem, exponent rules help break down and simplify the expression efficiently. For instance, applying the power of a power rule, \((m^{33})^2 = m^{66}\), allows simplification without manual repetition of terms.

Graphing Calculator

For problems involving large numerical calculations, a graphing calculator is an invaluable tool. It can handle difficult computations, making tasks such as squaring large numbers easier and more precise.

Using a graphing calculator, you can:

• Calculate high power values accurately (e.g., \(110592^2\))
• Graph polynomial expressions to visualize solutions
• Check computations with immediate accuracy
• Explore multiple forms of the same expression to find the simplest version

This empowers students to focus on understanding the algebra rather than getting bogged down by arithmetic. Using these features wisely helps connect theoretical understanding with practical applications.

Polynomial Expressions

In algebra, a polynomial expression consists of variables raised to a certain power and accompanied by coefficients. They play a crucial part in many algebra equations and require a solid understanding to manipulate effectively.

In the expression \(12230590464 m^{66}\), we recognize:

• The term \(m^{66}\) is the variable term raised to an exponent
• The coefficient \(12230590464\) as a numerical multiplier
• This expression is a monomial – a type of polynomial with one term

Polynomials can be added, subtracted, multiplied, or divided based on algebraic rules. It's important to simplify and express polynomial expressions clearly to reveal their underlying structure and make them easier to work with. Understanding these characteristics aids in effectively solving complex algebraic problems.