Problem 62 - Algebra 2

Question

Multiple Choice The expression 0.036\(m^{\frac{3}{4}}\) is used in the study of fluids. Which best represents the value of the expression for \(m=46 \times 10^{4} ?\) \(\begin{array}{lllll}{\text { A } 636} & {\text { B } 1460} & {\text { C } 1660} & {\text { D } 16,600}\end{array}\)

Step-by-Step Solution

Answer

The best representation for the given expression when \(m = 46 \times 10^{4}\) is 1460, so the correct answer is option B.

Step 1: Understand the given expression

The given expression is 0.036\(m^{\frac{3}{4}}\). Here \(m\) is a variable and the given value for it is \(46 \times 10^{4}\).

Step 2: Substitute \(m\) in the given expression

Substitute \(m = 46 \times 10^{4}\) into the given expression: \[0.036(46 \times 10^{4})^{\frac{3}{4}}\]

Step 3: Simplify the expression

First simplify the portion in the parenthesis, \((46 \times 10^{4})^{\frac{3}{4}}\). Next multiply the result by 0.036. Use a calculator for the calculations to get the final answer.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. The exponent tells us how many times the base number is multiplied by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. If \(n = 3\), then \(a^3 = a \times a \times a\).

For the given exercise, the exponent \(\frac{3}{4}\) means we are finding a value such that \(m\) is raised to this fractional power. Fractional exponents, such as \(m^{\frac{3}{4}}\), are interpreted as a combination of a root and an exponent.

• \(m^{\frac{1}{2}}\) is the square root.
• \(m^{\frac{1}{3}}\) is the cube root.
• \(m^{\frac{3}{4}}\) means the fourth root of \(m^3\).

For example, to calculate \((46 \times 10^4)^{\frac{3}{4}}\), first find the fourth root of \((46 \times 10^4)^3\). The concept of exponentiation is crucial for understanding how to manipulate and simplify expressions involving powers.

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a mathematical object or relationship. They do not include equality or inequality signs, which distinguishes them from equations.

In our exercise, the expression is \(0.036m^{\frac{3}{4}}\). Here, 0.036 is a constant that multiplies the variable expression \(m^{\frac{3}{4}}\). Expressions can be used to model real-world situations, such as fluid dynamics in this case, which often involves complex calculations with a variable like \(m\) given a specific context.

When working with expressions, you substitute values for the variables (like replacing \(m\) with \(46 \times 10^4\)), and then perform the operations indicated by the expression:

• Substitute the given value.
• Perform any arithmetic operations.
• Use algebraic rules to simplify the expression if possible.

Understanding expressions and how to manipulate them is key to problem solving in algebra.

Problem solving

Problem solving in mathematics involves identifying the goal, understanding the problem, planning a strategy, executing that strategy, and reviewing the solution. Each step is important for a thorough understanding and correct resolution of the problem.

For the expression \(0.036m^{\frac{3}{4}}\), the problem asks to find its value given \(m = 46 \times 10^4\). Here is a general approach for solving such problems:

• **Understand the problem:** Recognize all components of the expression and make sure you have the correct value to substitute for the variable.
• **Plan a strategy:** Decide whether to simplify first or substitute first. In our exercise, substitute \(m\) into \(m^{\frac{3}{4}}\).
• **Execute the strategy:** After substitution, calculate \((46 \times 10^4)^{\frac{3}{4}}\) and then multiply by 0.036. Often a calculator will be needed for accurate computations.
• **Review:** Double-check your calculations to ensure accuracy and verify that your result corresponds logically to the choices provided in the problem.

Mastering problem solving means leveraging these strategies to solve not just this exercise, but a wide range of mathematical problems.