Problem 73 - Algebra 2

Question

Solve each equation. $$ 25^{2 x+1}=144 $$

Step-by-Step Solution

Answer

Based on the solution steps, the solution to the given equation is \(x = 0\).

Step 1: Convert to the Same Base

Convert the number 25 and 144 into the same base. The number 5 can be used as a common base because both 25 and 144 have square roots that are integer numbers. So, we can rewrite the equation as follows: \( (5^2)^{2x+1} = (12^2)^1 \)

Step 2: Simplify the Equation

Simplify the equation using the properties of exponents, which states that \((a^m)^n = a^{mn}\): \( 5^{4x+2} = 12^2 \)

Step 3: Compare the Exponents and Solve for x

Since we have equal bases, we can set the exponents equal to each other: \(4x + 2 = 2\). Solving for x gives the solution: \(x = 0\)

Key Concepts

Exponents

Exponents are a mathematical notation that represents the number of times a base is multiplied by itself. Exponents are expressed as a small raised number placed to the right of a base number. For example, in the expression \(5^2\), the number 5 is the base, and 2 is the exponent, which means that 5 is multiplied by itself once to get 25.
Exponents are incredibly useful when dealing with large numbers or simplifying expressions, as they provide a concise way to express repeated multiplication.

• \(a^m\): \(a\) is the base and \(m\) is the exponent.
• Properties: Products \((a^m \times a^n = a^{m+n})\) and powers \(((a^m)^n = a^{mn})\).
• Solving equations often involves equating the exponents of common bases.

Understanding these properties helps in solving equations where terms need to be simplified or compared.

Integer Solution

An integer solution for an equation means that the solution for the unknown variable is a whole number. In our exercise, we solved for \(x\) and found that \(x = 0\), which is an integer.
Finding integer solutions is crucial in many mathematical problems and real-life applications such as counting, dividing objects into equal parts, and more. To determine if a solution is an integer, the steps in solving the equation must lead to a whole number.
Here are a few methods to ensure integer solutions:

• Simplify the equation using algebraic properties.
• Set conditions based on the initial problem constraints.
• Check the solution by substituting back into the original equation.

Integer solutions provide clarity and simplicity as they eliminate the complexity of fractional or decimal values.

Common Base

A common base is an essential concept when simplifying equations involving exponents. It involves expressing different terms with the same base to easily compare or solve them. In our example, we transformed the numbers 25 and 144 into terms with a common base by utilizing their square roots, which makes calculations easier.
Some reasons to use common bases are:

• Simplification: Reduces complex terms into simpler ones.
• Comparison: Allows for direct comparison of exponents to solve equations.
• Consistency: Maintains uniformity, which is helpful in both equations and inequalities involving exponents.

To find a common base, identify the smallest base that can express each term as some power of that base. This often involves factorization or recognizing the structure of the terms involved. Using a common base makes equations more manageable and leads directly to solving for the unknown variables.