Problem 106 - Elementary and Intermediate Algebra

Question

Use the associative property of multiplication to find each product. $$ \left(-\frac{5}{16} \cdot \frac{1}{7}\right) 7 $$

Step-by-Step Solution

Answer

The product is \(-\frac{5}{16}\).

Step 1: Identify the Expression

The given expression is \(-\frac{5}{16} \cdot \frac{1}{7} \cdot 7\). We need to rearrange this expression using the associative property of multiplication.

Step 2: Apply the Associative Property

The associative property of multiplication states that \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). Therefore, we can rearrange our expression as \(-\frac{5}{16} \cdot \left(\frac{1}{7} \cdot 7\right)\).

Step 3: Simplify the Inner Multiplication

Compute \(\frac{1}{7} \cdot 7\). Since multiplying a fraction by its reciprocal results in 1, we have \(\frac{1}{7} \cdot 7 = 1\).

Step 4: Complete the Multiplication

Substitute back into the expression: \(-\frac{5}{16} \cdot 1 = -\frac{5}{16}\). Therefore, the product is \(-\frac{5}{16}\).

Key Concepts

Fractions Multiplication

When multiplying fractions, understanding the basic steps can simplify the process greatly. Multiplication of fractions involves two main steps:

• Multiply the numerators (the top numbers).
• Multiply the denominators (the bottom numbers).

For instance, to multiply \( -\frac{5}{16} \) and \( \frac{1}{7} \), you simply find the product of the numerators and the product of the denominators:

• Numerator: \(-5 \times 1 = -5\)
• Denominator: \(16 \times 7 = 112\)

This gives us the fraction \( -\frac{5}{112} \).
Remember that if one of the fractions is negative, the resulting product will also be negative. Fractions multiplication is a straightforward process, but simplifying results may also be necessary in some cases.

Rearranging Expressions

Rearranging expressions allows us to use arithmetic properties more effectively to simplify mathematical tasks. The associative property of multiplication is particularly useful here. This property tells us that the way in which numbers are grouped in multiplication does not change their product:
\[(a \cdot b) \cdot c = a \cdot (b \cdot c)\]This means we can change the grouping of numbers based on convenience. Take the expression \(-\frac{5}{16} \cdot \frac{1}{7} \cdot 7\). Applying the associative property, we regroup this as \(-\frac{5}{16} \cdot (\frac{1}{7} \cdot 7)\). This allows us to first easily simplify \(\frac{1}{7} \cdot 7\), knowing the result will conveniently be 1.

Multiplying with Reciprocals

Multiplying a number by its reciprocal always yields 1. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). When a fraction and its reciprocal are multiplied, the numerator and denominator cancel each other out, leaving you with 1:
\[\frac{a}{b} \cdot \frac{b}{a} = \frac{ab}{ba} = 1\]In our specific problem, \( \frac{1}{7} \) is multiplied by its reciprocal, 7, which simplifies directly to 1.
This step significantly simplifies the calculation process, turning a complex multiplication problem into something much more manageable: \(-\frac{5}{16} \cdot 1 = -\frac{5}{16}\). Recognizing and using reciprocals can be an effective strategy to simplify expressions quickly.