Problem 63 - 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 \(m=46 \times 10^{4} ?\) A 636 B 1460 C 1660 D \(16,600\)

Step-by-Step Solution

Answer

The best representation for the value of the expression when \(m = 46 \times 10^{4}\) is not listed among the options A, B, C, and D. The closest answer is A. The value calculated is \(463\)

Step 1: Substitute \(m\) in the expression

Replace \(m\) with its given value in the expression. This gives: 0.036 \((46 \times 10^{4})^{\frac{3}{4}}\)

Step 2: Calculate the power first

According to the rule of order of operations (BIDMAS/BODMAS), powers should be calculated before multiplication. Therefore, calculate \((46 \times 10^{4})^{\frac{3}{4}}\) first.

Step 3: Simplify the power

To simplify \((46 \times 10^{4})^{\frac{3}{4}}\), calculate \(46^{\frac{3}{4}}\) and \((10^{4})^{\frac{3}{4}}\) separately. \((10^{4})^{\frac{3}{4}}\) simplifies to \(10^{3} = 1000\), and \(46^{\frac{3}{4}} = 12.884\), rounding to three decimal places. Multiply these two results together to get \(12.884 \times 1000 = 12884\)

Step 4: Complete the multiplication

Now, multiply the result from step 3 with 0.036. Therefore, \(0.036 \times 12884 = 463.824\)

Step 5: Round to the nearest whole number

The options given in the exercise are in whole numbers. Therefore, we round the answer from step 4 to the nearest whole number. Hence, \(463.824\) rounds down to \(463\)

Key Concepts

Order of Operations

Understanding the order of operations is crucial to correctly solving mathematical expressions. Following certain rules ensures that calculations are carried out in the correct sequence. Commonly remembered by the acronym BIDMAS/BODMAS, which stands for Brackets, Indices (or Orders), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

These steps should be applied in the specified order to avoid errors. For instance, in the expression \(0.036(m^{\frac{3}{4}})\), it is important to first resolve any powers or exponents, like \((m^{\frac{3}{4}})\), before proceeding with multiplication. This type of structure ensures clarity and accuracy in evaluations.

Exponents and Powers

Exponents, also known as powers, denote the number of times a number (the base) is multiplied by itself. In mathematical notation, it is shown as \(m^{n}\), where \(m\) is the base and \(n\) is the exponent.

Powers are calculated before performing other operations in expressions due to their high precedence in order of operations. For the expression \((46 \times 10^{4})^{\frac{3}{4}}\), both bases \(46\) and \(10^{4}\) are raised to the power of \(\frac{3}{4}\). Thus, separately calculating \(46^{\frac{3}{4}} \approx 12.884\) and \((10^{4})^{\frac{3}{4}} = 1000\), and then combining these gives a significant value that affects further calculations.

Mastering exponents is key as they often appear in scientific and engineering contexts, like fluid studies, where such expressions are used to model complex situations.

Numerical Substitution

Numerical substitution involves replacing a variable with a specific number to simplify and solve an expression. This process is essential for evaluating expressions where specific values are provided for the variables.

In the given exercise, the expression \(0.036(m^{\frac{3}{4}})\) features the variable \(m\) which is given as \(46 \times 10^{4}\). Substituting \(m\) with this value transforms the expression into a solvable form: \(0.036 (46 \times 10^{4})^{\frac{3}{4}}\).

By systematically substituting and then solving the expression step-by-step, ensuring that each calculated result is accurate, students can effectively determine the correct outcome, such as evaluating the expression to get the rounded answer of 463.824 which simplifies to 463. Mastery of substitution techniques is invaluable in ensuring precision in mathematical problem-solving.