Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation for \(x\). $$(2.1)^{x+2}=(2.1)^{5}$$

Step-by-Step Solution

Answer

The solution to the equation \((2.1)^{x+2} = (2.1)^{5}\) is \(x = 3\).

Step 1: Equating the exponents

Since the bases are the same, we can equate the exponents: \(x + 2 = 5\)

Step 2: Solving for x

Now we need to solve the equation for x: \(x + 2 - 2 = 5 - 2\) \(x = 3\) Therefore, the solution to the equation \((2.1)^{x+2} = (2.1)^{5}\) is \(x = 3\).

Key Concepts

Algebra

Algebra is a fundamental part of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. It's like the language of mathematics. In algebra, solving equations means finding the values for the unknown variables that make the equation true. For instance, in the equation \( (2.1)^{x+2} = (2.1)^5 \), we are tasked with determining the value of \( x \) that satisfies this equation. A crucial concept in algebra is understanding how to manipulate equations to isolate the variable. This involves using operations like addition, subtraction, multiplication, and division. In our example equation, by recognizing that the bases of the expressions on both sides are the same, we directly equate the exponents. This allows us to greatly simplify our process to find the value of \( x \). Mastering algebra provides the foundation for more advanced mathematical concepts and problem-solving abilities. It is widely used across different fields, from engineering to economics. It's all about identifying patterns and setting up equations to model and solve real-world problems.

Mathematical Exponents

Exponents are a way to represent numbers in terms of powers. They indicate how many times a number, known as the base, is multiplied by itself. For instance, in \((2.1)^{x+2}\), \(2.1\) is the base, and \(x+2\) is the exponent.When working with exponents, there are several rules that can simplify calculations:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents.
• Power of a Power Rule: When raising a power to another power, multiply the exponents.

In the original problem, since the bases \( (2.1) \) on both sides are the same, we can set the exponents equal to each other. This makes it extremely straightforward to solve the equation. Understanding how to manipulate and work with exponents is essential for simplifying expressions and solving equations efficiently in algebra.

Equation Solving

Equation solving is all about finding the unknown values that satisfy a given equation. It's a fundamental skill in algebra and involves a sequence of logical steps. The key is to apply operations that maintain the equality of the equation while isolating the unknown variable.In our example \( (2.1)^{x+2} = (2.1)^5 \), solving the equation involved recognizing that both sides have the same base. This lets us equate the exponents directly, simplifying \( x+2 = 5 \) to \( x = 3 \).When solving such equations, you should:

• Ensure to check and simplify both sides of the equation if necessary.
• Use algebraic operations to isolate the variable. Here, we subtract 2 from both sides to solve for \( x \).
• Always verify your solution by substituting it back into the original equation to ensure both sides equal.

Equation solving allows us to derive values that are consistent and balanced, and is an integral part of mathematical problem-solving and critical thinking skills.