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Chapter 0: Problem 3

Determine how many different values can arise by inserting one pair of parentheses into the given expression. $$ 6+3 \cdot 4+5 \cdot 2 $$

Short Answer

Expert verified
There are \(\boxed{3}\) different values that can arise by inserting one pair of parentheses into the given expression: \(46, 60\), and \(28\).

Step by step solution

01

Analyze the expression

The given expression is: \( 6+3 \cdot 4+5 \cdot 2 \) We have three addition operations and two multiplication operations. We will insert parentheses around different pairs of operations to find all possible combinations.
02

Insert parentheses around addition operations

First, let's consider inserting parentheses around addition operations in the expression. We have three possibilities: 1) \((6+3) \cdot 4+5 \cdot 2\) 2) \( 6+3 \cdot (4+5) \cdot 2\) 3) \( 6+3 \cdot 4+(5 \cdot 2)\) Now, we will compute the value of each expression: 1) \((6+3) \cdot 4+5 \cdot 2 = 9 \cdot 4 + 10 = 36+10=46\) 2) \( 6+3 \cdot (4+5) \cdot 2 = 6+3 \cdot 9 \cdot 2 = 6+27 \cdot 2 = 6+54 = 60\) 3) \( 6+3 \cdot 4+(5 \cdot 2) = 6+12+10 = 28\) So we have three different values: \(46, 60, \text{and } 28\).
03

Insert parentheses around multiplication operations

Now let's insert parentheses around multiplication operations in the expression. We have two possibilities: 1) \( 6+(3 \cdot 4)+5 \cdot 2\) 2) \( 6+3 \cdot 4+5 \cdot (2)\) Now we will compute the value of each expression: 1) \( 6+(3 \cdot 4)+5 \cdot 2 = 6+12+10 = 28\) 2) \( 6+3 \cdot 4+5 \cdot (2) - 6+12+10 = 28\) As we can see, these new combinations produce the same values as in step 2 (28). So, the result doesn't change.
04

Count the different values

Finally, we will count the different values achieved by inserting the parentheses: 1) \(46\) 2) \(60\) 3) \(28\) Hence, we have \(\boxed{3}\) different values that can arise by inserting one pair of parentheses into the given expression.

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Most popular questions from this chapter

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(31-40,\) write each intersection as a single interval. $$ (-\infty, 4) \cap(-2,6] $$

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(31-40,\) write each intersection as a single interval. $$ [-8,-3) \cap[-6,-1) $$

In Exercises \(19-30,\) write each set as an interval or as a union of two intervals. $$ \left\\{x:|4 x-3|<\frac{1}{5}\right\\} $$

In Exercises \(7-16,\) write each union as a single interval. $$ (-\infty,-6] \cup(-8,12) $$

Expand the given expression $$ (b-3)(b+3)\left(b^{2}+9\right) $$
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