Problem 3 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 2^{2 x}=16 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

The given equation is \(2^{2x} = 16\). Our objective is to solve for \(x\). We need to express both sides with the same base if possible to simplify the equation.

Step 2: Express 16 as a Power of 2

We recognize that 16 can be written as a power of 2. Specifically, \(16 = 2^4\). This will allow us to rewrite the equation with the same base on both sides.

Step 3: Rewrite the Equation

Substitute \(16\) in the equation with its equivalent power of 2. Now the equation is \(2^{2x} = 2^4\).

Step 4: Set the Exponents Equal

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

Step 5: Solve for x

Divide both sides of the equation by 2 to isolate \(x\): \(x = \frac{4}{2} = 2\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the value of the unknown variable that satisfies the equation. In our example, we want to solve the equation \(2^{2x} = 16\). Here, solving means determining what value \(x\) should take so that both sides of the equation are equal.
When solving equations, especially those involving exponents, it's crucial to first simplify the situation if possible.

• The first step is to rewrite the equation in such a way that allows you to easily identify equal expressions or use algebra to isolate the variable.
• Look for opportunities to work with similar bases or expressions on both sides of the equation, which helps in further simplification.

In this case, expressing 16 as a power of 2 helps us deal directly with the same base on both sides, enabling an easier comparison of exponents.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. In the expression \(a^b\), \(a\) is the base and \(b\) is the exponent, indicating that \(a\) should be multiplied by itself \(b\) times.
Exponents follow specific rules that help in simplifying expressions:

• Power of a Product: \((ab)^n = a^n \cdot b^n\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)

In our equation, \(2^{2x}=16\), knowing that 16 can be written as \(2^4\) is crucial because it allows comparison based on the rules of exponents. Recognizing similar expressions in terms of powers simplifies our path to finding the solution for \(x\).

Powers of Numbers

Understanding powers of numbers is key when dealing with equations like \(2^{2x} = 16\). This concept revolves around multiplying a number by itself several times. The powers correspond to the number of times this multiplication is carried out.
For the number 2:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)

Each increase in the exponent results in rapid growth due to the doubling nature of this sequence. Recognizing that 16 is the result of \(2^4\) allowed us to make the sides of our equation match up in terms of their bases.
This fundamental understanding helps us replace 16 in the original equation and eventually solve for \(x\) step-by-step. By transforming the equation into \(2^{2x} = 2^4\), setting the exponents equal is justified because the bases, being identical, imply that the exponents themselves must equate to keep the equation true.