Problem 61 - Algebra 2

Question

Solve each equation or inequality. Check your solution. \(2^{2 x}=\frac{1}{8}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{-3}{2}\).

Step 1: Understand the Equation

The equation given is \(2^{2x} = \frac{1}{8}\). To solve it, recognize that \(\frac{1}{8}\) can be rewritten as a power of 2.

Step 2: Rewrite the Right Side

Since \(8 = 2^3\), the expression \(\frac{1}{8}\) can be rewritten as \(2^{-3}\). Thus, the equation becomes \(2^{2x} = 2^{-3}\).

Step 3: Set Exponents Equal

Since the bases on both sides of the equation are equal, the exponents must also be equal. Therefore, set the exponents equal: \(2x = -3\).

Step 4: Solve for x

To find \(x\), divide both sides of the equation by 2, resulting in \(x = \frac{-3}{2}\).

Step 5: Check the Solution

To verify the solution, substitute \(x = \frac{-3}{2}\) back into the original equation: \(2^{2(-3/2)} = 2^{-3}\). Both sides of the equation equal \(\frac{1}{8}\), confirming that \(x = \frac{-3}{2}\) is correct.

Key Concepts

Solving Equations

Solving equations is a key skill in mathematics. It involves finding the unknown value that makes the equation true. When solving equations like \(2^{2x} = \frac{1}{8}\), the first step is to understand both sides of the equation. Here, recognizing that the number \(8\) can be expressed as \(2^3\) is critical. By doing so, we see that \(\frac{1}{8} = 2^{-3}\).
This insight allows us to transform the equation into a simpler form: \(2^{2x} = 2^{-3}\). With identical bases on both sides, solving the equation becomes easier: we can directly set the exponents equal since \(a^b = a^c\) implies \(b = c\) for \(a > 0\).
Thus, we simplify to \(2x = -3\), providing a straightforward step to solve for \(x\). Dividing by 2 gives us the solution \(x = -\frac{3}{2}\).

Powers of Numbers

Understanding powers of numbers and exponential expressions is essential in solving many types of equations. In exponential equations, a number (called the base) is raised to a power. For instance, in \(2^{2x}\), the base is \(2\) and the exponent is \(2x\).
This concept is crucial because it often lets us rewrite numbers in different forms. The number \(8\), for instance, can be rewritten using powers of \(2\) as \(8 = 2^3\). Consequently, \(\frac{1}{8} = 2^{-3}\), which shows how exponents can be negative, reflecting division by the base rather than multiplication.
The ability to re-express numbers this way is powerful. It not only lets us solve equations where bases are common but also allows us to understand how growth and decay work in exponential contexts, such as doubling, halving, and more.

Checking Solutions

Once a potential solution is found, it is always important to verify it. Checking solutions ensures we have accurately solved the equation. To check, we substitute the solution back into the original equation.

• In this exercise, after finding \(x = -\frac{3}{2}\), we substitute it back into \(2^{2x} = \frac{1}{8}\).
• This involves calculating \(2^{2(-\frac{3}{2})} = 2^{-3}\), which indeed is equal to \(\frac{1}{8}\).

This confirmation step, though brief, is crucial. It affirms the correctness of our solution and gives us confidence in our problem-solving process. Always remember to check because small mistakes can happen in previous steps, and this act is your safety net against simple errors.