Problem 55 - Thinking Mathematically

Question

a^{4}=a^{10} \quad a^{3} \\# a^{2}=a^{7} \quad a^{5} \\# a^{3}=a^{11} . $$ Select the equation that describes the pattern… # Study the pattern in these examples: $$ a^{2} \\# a^{4}=a^{10} \quad a^{3} \\# a^{2}=a^{7} \quad a^{5} \\# a^{3}=a^{11} . $$ Select the equation that describes the pattern. a. \(a^{x} \\# a^{y}=a^{2 x+y}\) b. \(a^{x} \\# a^{y}=a^{x+2 y}\) c. \(a^{x} \\# a^{y}=a^{x+y+4}\) d. \(a^{x} \\# a^{y}=a^{x y+2}\)

Step-by-Step Solution

Answer

The equation that describes the pattern is option (b), \(a^{x} \# a^{y} = a^{x+2 y}\).

Step 1: Identify the Pattern

First, let's examine the initial given equations. Notice that the exponents on the right side of each equation are always larger than the exponents on the left side. Let's see if we can identify a pattern.

Step 2: Compare the Equations

Compare the given equations to the options. For the first equation \(a^{2} \# a^{4} = a^{10}\), the exponents 2 and 4 on the left add up to 6. The difference between the resulting exponent 10 and the sum of the exponents on the left side of the equation (6) is 4.

Step 3: Apply Pattern to Other Equations

Now let's apply the same principle to the other given equations. For \(a^{3} \# a^{2} = a^{7}\), the exponents 3 and 2 add up to 5, subtract this from 7 (the resulting exponent), we again get 2. Similarly, for \(a^{5} \# a^{3} = a^{11}\), adding the exponents 5 and 3 will give 8, subtracting 8 from 11 also gives us 3.

Step 4: Select the Equation that Matches the Pattern

Now, the equation that matches the pattern we've identified is \(a^{x} \# a^{y} = a^{x+y+2}\). So, the answer is option (b).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It serves as a unifying thread in almost all of mathematics. When working with equations in algebra, we often deal with unknown values, represented by variables like \(a\), \(x\), or \(y\). These variables allow us to represent a general case or pattern which can be solved to find specific solutions.

In algebraic expressions and equations, we manipulate these symbols according to rules that resemble arithmetic. For example, we can add, subtract, multiply, or divide both sides of an equation to solve for a variable. In the case of our given problem, we are examining patterns in equations that involve exponentiation. The goal is to identify how these exponents interact to discern a broader rule or equation that describes their behavior.

Exponents

Exponents tell us how many times to multiply a number by itself. For instance, \(a^3\) means \(a\) multiplied by itself three times: \(a \times a \times a\). Exponents are a compact way of expressing repeated multiplication. They play a key role in algebraic equations, especially when understanding patterns and solving problems involving powers.

In the context of our exercise, exponents are used to describe patterns. When you see an expression like \(a^x \# a^y = a^{x+y+2}\), it indicates that we're combining two exponential expressions in a specific way. This pattern shows that after adding the exponents \(x\) and \(y\), there's an added increment, here 2, to the result. Recognizing such patterns and rules can greatly simplify analyzing complex equations, enabling us to predict or calculate outcomes based on the given operations.

Equation Solving

Equation solving in algebra involves finding the value of the variable that makes the equation true. It is a fundamental skill in mathematics that unlocks the solution to countless problems. Solving equations with exponents requires a solid understanding of how to manipulate both sides of the equation efficiently.

In the problem we're addressing, solving involves identifying how different parts of the given expression contribute to the pattern or the equation as a whole. By looking at the relationships among the exponents, we glean insights into how they transform.

• We identify patterns by comparing the relationships in each presented equation.
• Recognizing that the right side of the equation always equaled the sum of left side exponents plus a constant.
• Ultimately, identifying which given option accurately describes this observed pattern.

In our case, the solution was found by verifying each option with the pattern derived from the given examples, leading us to see that the correct one is \(a^{x} \# a^{y} = a^{x+y+2}\). This process, intricate yet logical, illustrates how methodical equation solving can yield the key to understanding complex patterns.