Problem 22 - Algebra for College Students

Question

Solve each of the equations. $$ \left(2^{2 x-1}\right)\left(2^{x+2}\right)=32 \quad\left\\{\frac{4}{3}\right\\} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{3}\).

Step 1: Simplify the Left Side Using Exponent Rules

We are given the equation \((2^{2x-1})(2^{x+2}) = 32\frac{4}{3}\). Let's simplify the left side first. Using the property of exponents \(a^m \cdot a^n = a^{m+n}\), we combine the exponents of the two terms with base 2: \(2^{2x-1+x+2} = 2^{3x+1}\).

Step 2: Simplify the Right Side

The given right side is \(32\frac{4}{3}\). Let's express this in a single fraction: \(32\frac{4}{3} = \frac{32 \times 3 + 4}{3} = \frac{96+4}{3} = \frac{100}{3}\).

Step 3: Express \(32\) as a Power of 2 and Simplify

Express 32 as \(2^5\). So the original problem on the right-hand side becomes: \(2^5\frac{4}{3}\), which is consistent with the expression \(\frac{100}{3}\).

Step 4: Set Exponents Equal and Solve for \(x\)

Since we know that \(2^{3x+1}\) must equal \(2^5\frac{4}{3}\), specifically focusing on the integer exponents side, we equate the exponents: \(3x+1 = 5\), noting that powers must adhere to equality even in mixed numbers when reduced. Solving for \(x\), \(3x = 5 - 1\), which gives \(3x = 4\). Thus, \(x = \frac{4}{3}\).

Key Concepts

Properties of Exponents

In mathematics, understanding the properties of exponents is crucial when tackling exponential equations. These properties help us simplify and solve equations by applying specific rules about how exponents interact. Here are some essential rules:

• Product of Powers: This property states that when two powers with the same base are multiplied, you can add their exponents. Mathematically, this is expressed as \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another exponent, multiply the exponents. In other words, \((a^m)^n = a^{mn}\).
• Power of a Product: When a product is raised to an exponent, you can distribute the exponent to each factor within the product: \((ab)^m = a^m \cdot b^m\).

These rules are instrumental, especially when we're dealing with equations where exponents need to be manipulated for simplification or comparison. For example, in the context of our exercise, we used the Product of Powers property to combine \(2^{2x-1}\) and \(2^{x+2}\) into \(2^{3x+1}\). This simplification is a key step in making the equation easier to solve.

Equation Simplification

Simplifying an equation is about making it easier to understand and solve. This often involves reducing the equation's complexity by combining like terms or converting expressions into simpler forms. Let's look at how we can simplify various parts of an equation:

• Combine Like Terms: When terms have the same variables to the same power, you can add or subtract them as needed.
• Simplify Expressions: Use algebraic properties, such as distribution or factoring, to simplify complex expressions.

In terms of our original exercise, the left-hand side \((2^{2x-1})(2^{x+2})\) was simplified using properties of exponents to create a single term \(2^{3x+1}\). Similarly, the right-hand side involved simplifying \(32 \frac{4}{3}\) into the single fraction \(\frac{100}{3}\). This approach makes the equation more direct and sets up for the next steps in solving it.

Fraction Operations

Fractions can sometimes complicate an equation, but by understanding basic operations with fractions, you can efficiently maneuver through them. Here's how you can simplify fractional expressions:

• Addition of Fractions: To add fractions, find a common denominator, adjust the numerators accordingly, and then combine them.
• Multiplication of Fractions: Multiply the numerators together and the denominators together separately.
• Conversion: Mixed numbers can be converted to improper fractions for ease of calculation. For example, \(32\frac{4}{3}\) can be converted by \(32 \times 3 + 4 = 100\), resulting in \(\frac{100}{3}\).

Applying these operations, as in our exercise, allows us to simplify the expression effectively, leading to a cleaner equation that is less daunting to solve.

Exponent Equality

The concept of exponent equality is vital when solving equations that involve exponents. This principle asserts that if the bases are the same, then the exponents can be set equal to each other. This strategy simplifies the process of finding unknown variables.Let's break this down further:

• If \(a^m = a^n\), then it must follow that \(m = n\).
• This is assuming \(a\), the base, is not equal to zero.

In the given exercise, once the expression was simplified so that both sides had the base 2, we could set the exponents \(3x+1\) and 5 equal to each other. This simplification allows us to directly solve for \(x\) without needing to handle the base further. By applying exponent equality effectively, solving exponential equations becomes a systematic and simplified process.