Problem 4 - Algebra for College Students

Question

Solve each of the equations. $$ 2^{2 x}=16 $$

Step-by-Step Solution

Answer

\(x = 2\).

Step 1: Understand the Equation

The given equation is \(2^{2x} = 16\). Our goal is to solve for \(x\). Notice that the left-hand side is an exponential expression with base 2.

Step 2: Express Both Sides with the Same Base

Recognize that 16 can be expressed as a power of 2. Specifically, \(16 = 2^4\). So we rewrite the equation as \(2^{2x} = 2^4\).

Step 3: Set the Exponents Equal

Since the bases are the same, we set the exponents equal to each other: \(2x = 4\).

Step 4: Solve for x

To find \(x\), divide both sides of the equation by 2: \(x = \frac{4}{2}\).

Step 5: Simplify

Simplify \(x = \frac{4}{2}\) to get \(x = 2\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in mathematics that involves finding the values of variables that make the equation true. In the exercise provided, we aim to solve the equation \(2^{2x} = 16\). The process of solving an equation typically involves the following steps:

• Identify the type of equation you are dealing with. In this case, it is an exponential equation.
• Express both sides of the equation in a form that will allow you to compare coefficients, such as converting numbers to the same base.
• Set the equations of the same base equal to each other to isolate the variable.
• Solve for the unknown variable.

Working through an equation step by step helps us understand how changes to one part of the equation affect the solution. By breaking down each step, solving equations becomes a straightforward process, ensuring that all work is accurate and understandable.
Exponential equations, like the one in our exercise, can often be easily solved by leveraging properties of exponents.

Exponents

Exponents, also known as powers, are a way to express repeated multiplication of a number by itself. For instance, \(2^4\) means we multiply 2 by itself four times, equating to 16. Understanding the properties of exponents simplifies many equations and mathematical expressions. Here are some key concepts about exponents:

• An exponent indicates how many times the base is used as a factor.
• The expression \(a^b\) is called an exponential expression, where \(a\) is the base and \(b\) is the exponent.
• When the bases are the same, you can set the exponents equal if you have an equality like \(a^c = a^d\).

By converting both sides of our equation to have the same base, we can use the property that states if the bases are equal, the exponents must also be equal when the overall expression equals one another.
This foundational knowledge about exponents is useful in solving exponential equations, streamlining the process of polynomial simplifications and solutions.

Base of Power

The concept of the base of a power is central to understanding exponential equations. The base is the number that is being raised to a power by the exponent. In our exercise, the number 2 is referred to as the "base". Understanding this allows us to manipulate equations effectively:

• The base of a power remains constant in any identical conversion of the expression.
• Recognizing equivalent bases on both sides of an equation lets us set the exponents equal.
• Rewriting numbers in terms of a common base can often simplify or solve equations more efficiently.

In our solved equation \(2^{2x} = 2^4\), both sides of the equation have the base 2, allowing us to set \(2x\) equal to 4 directly. This principle is powerful and common when handling exponential equations, greatly simplifying the solving process.
Practicing identifying and using the base of power enhances your ability to tackle a wide range of mathematical problems, from simple algebraic equations to more complex calculus-based scenarios.