Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\(\frac{2^{6}}{2^{3}}\)

Step-by-Step Solution

Answer

The simplest form of the expression \(\frac{2^{6}}{2^{3}}\) is 8.

Step 1: Identify the bases and exponents

In the expression, \( \frac{{2^{6}}}{{2^{3}}}\), the base is 2 and the exponents are 6 and 3 respectively.

Step 2: Apply the rule of exponents for division

When dividing terms with the same base, we can subtract the exponents. So, \( \frac{{2^{6}}}{{2^{3}}} = 2^{6-3}\)

Step 3: Calculate the answer

Now calculate \(2^{6-3}\) which will be \(2^{3}\), which equals to 8

Key Concepts

Bases and Exponents

In mathematics, understanding bases and exponents forms a foundation for working with powers and expressions. The base refers to the number that is multiplied by itself a certain number of times, which is determined by the exponent. In our example, the base is 2 in both the numerator and the denominator. The exponent indicates the number of times the base is used as a factor. For example, in the expression \(2^6\), 2 is the base and 6 is the exponent, meaning you multiply 2 by itself six times (\(2 \times 2 \times 2 \times 2 \times 2 \times 2\)). Similarly, in \(2^3\), 2 is used as a factor three times. Understanding these concepts:

• The base is the number you repeatedly multiply.
• The exponent shows how many times the base is used in the multiplication.

Grasping these aspects of bases and exponents makes it easier to understand and simplify expressions involving powers.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form by canceling common factors in the numerator and denominator. For fractional expressions with exponents like \(\frac{2^{6}}{2^{3}}\), this process is slightly different yet straightforward. Here, instead of cancelling common numerical factors, we use the properties of exponents.The expression \(\frac{2^{6}}{2^{3}}\) simplifies via the rules of exponents, specifically by subtracting exponents. Although it might initially seem complex, the key to simplifying is focusing on the bases being identical. By rearranging the exponents, you effectively reduce the fraction to its simplest form. You can visualize it as reducing the operations needed. This process takes the form of, \(2^{6-3}\), simplifying neatly to \(2^3\), or 8.Let's remember:

• Simplifying means breaking down a complex form to a simpler one.
• For fractional exponents, match the bases and then simplify by exponent rules.

Rules of Exponents

The rules of exponents provide a toolkit for working with powers in mathematical expressions. When tackling expressions such as \(\frac{2^{6}}{2^{3}}\), one key rule is employed: **Division of Powers with the Same Base:** When dividing exponential terms that have the same base, simply subtract the exponent of the denominator from the exponent of the numerator. This principle is represented as \(a^m/a^n = a^{m-n}\). Applying this rule:

• Check if the bases are the same.
• Subtract the lower exponent from the higher exponent.

This results in \(2^{6-3}\), which leads to \(2^3\), equaling 8.Remembering these fundamental rules allows for simplification of exponential expressions and rational calculations, even when numbers seem large or complex.