Problem 1 - Algebra for College Students

Question

Solve each of the equations. $$ 2^{x}=64 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Understand the Problem

The equation we are working with is \(2^x = 64\). This means that some power of 2 equals 64.

Step 2: Rewrite 64 as a Power of 2

Recognize that 64 is a power of 2. We can express 64 as \(2^6\) since \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).

Step 3: Set Exponents Equal

Since \(2^x = 64\) and we have rewritten 64 as \(2^6\), we can equate the exponents: \(x = 6\).

Step 4: Solution

Based on the previous step, the solution to the equation \(2^x = 64\) is \(x = 6\).

Key Concepts

Powers of Numbers

Powers of numbers refer to expressions that involve a base number raised to an exponent. This exponent indicates how many times the base number is multiplied by itself. For example, in the expression \(2^6\), the number 2 is the base, and 6 is the exponent. This tells us to multiply 2 by itself 6 times. Here’s how it works:

• \(2^1 = 2\)
• \(2^2 = 2 \times 2 = 4\)
• \(2^3 = 2 \times 2 \times 2 = 8\)
• \(2^4 = 2 \times 2 \times 2 \times 2 = 16\)
• \(2^5 = 32\)
• \(2^6 = 64\)

Understanding powers of numbers helps simplify problems and recognize patterns, such as when determining what power of a number results in a specific value.
Recognizing that \(64\) is \(2^6\) shows a grasp of powers of numbers and confirms why transformations in equations help solve them.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. For exponential equations, like \(2^x=64\), which involve powers, isolating the variable x is a crucial step. Here’s how it typically works:

• Identify the base and express both sides of the equation in terms of this base if possible.
• Set the exponents equal to each other, once both sides have the same base.
• Solve for the variable.

In our exercise, we recognized that the equation had the form \(2^x = 64\). By rewriting 64 as the power \(2^6\), we brought both sides to the same base. This allowed us to simplify the task to finding \(x\).
By equating the exponents, we found \(x = 6\).
This approach makes it simpler to solve because you are comparing exponents instead of computing with large numbers.

Mathematical Notation

Mathematical notation is a language used to represent mathematical ideas and concepts clearly and concisely. In solving the equation \(2^x = 64\), understanding this notation is crucial.
Here are key components of mathematical notation:

• Exponents: Denote repeated multiplication of a base number, as in \(2^x\).
• Equality: The equal sign \(=\) indicates that two expressions are the same in value.
• Variables: Letters like \(x\) that stand for unknown values to find.

Knowing how to read and manipulate this notation allows you to transform and solve expressions and equations.
In our example, replacing \(64\) with \(2^6\) leveraged mathematical notation to simplify the problem and direct us towards the solution: \(x = 6\). Understanding these symbols can enhance problem-solving efficiency and accuracy.