Problem 21 - College Algebra

Question

Evaluate each exponential expression. $$ \frac{2^{3}}{2^{7}} $$

Step-by-Step Solution

Answer

The expression \( \frac{2^{3}}{2^{7}}\) evaluates to \(\frac{1}{16}\)

Step 1: Identifying the Base and Exponents

Here, 2 is the base for both expressions. 3 is the exponent in the numerator and 7 is the exponent in the denominator.

Step 2: Rule of Exponents for Divisions

According to the rules of exponents for divisions, \(\frac{a^{m}}{a^{n}}\) equals \(a^{m-n}\). So, we can rewrite the given expression as \(2^{3-7}\).

Step 3: Subtract Exponents

Subtracting the exponents, we get \(2^{-4}\).

Step 4: Definition of Negative Exponent

According to the definition of a negative exponent, \(a^{-m} = \frac{1}{a^{m}}\). So, \(2^{-4} = \frac{1}{2^{4}}\).

Step 5: Solve Exponential Expression

Evaluating \(2^4\) gives 16. So, the final value is \(\frac{1}{16}\).

Key Concepts

Understanding Exponents

An exponent is a little number you see in the upper right corner of a base number, like in the expression \(2^3\). Here, 2 is the base, and 3 is the exponent. The exponent tells you how many times to multiply the base by itself. So, \(2^3\) means 2 multiplied by itself 3 times: \(2 \times 2 \times 2\). This gives us 8. Exponents are a shorthand way to express repeated multiplication, making large calculations more manageable. When you see expressions like this, remember:

• The base is the number you'll multiply.
• The exponent tells you how many times to use the base in a multiplication.
• The whole expression represents the repeated multiplication of the base.

Understanding exponents allows you to simplify and evaluate exponential expressions easily.

Grasping Negative Exponents

Negative exponents might seem tricky at first, but they're quite intuitive. A negative exponent indicates that you need to take the reciprocal of the base raised to the corresponding positive exponent. For instance, \(a^{-m}\) is equal to \(\frac{1}{a^m}\). So, if you have \(2^{-4}\), it actually means \(\frac{1}{2^4}\). With negative exponents, keep these points in mind:

• The negative sign flips the base into a fraction, from the numerator to the denominator.
• This transformation makes calculations simpler when combined with other numbers.

Understanding negative exponents will help you evaluate expressions even when the exponent isn't positive.

Mastering Rules of Exponents

The rules of exponents provide a set of clear, consistent guidelines for handling exponential expressions. They help simplify calculations, especially with more complex relationships between the bases and exponents. One crucial rule involves division, known as the Quotient of Powers Rule, which states: \(\frac{a^m}{a^n} = a^{m-n}\). This means when dividing like bases, you subtract the exponents. Here are some key rules to remember:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^m = a^m \cdot b^m\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Zero Exponent: Any number to the zero power is 1, \(a^0 = 1\) (as long as \(a\) is not zero).

Familiarity with these rules will enhance your ability to simplify and solve a wide range of exponential expressions.

Evaluating Powers Effectively

Evaluating powers involves simplifying exponential expressions to reach a numeric value. It requires a step-by-step approach and understanding of exponents and their rules. Following the example problem, first identify and apply the relevant rules. For \(\frac{2^3}{2^7}\), recognize the base and exponents, then apply the Quotient of Powers rule:1. Rearrange as \(2^{3-7}\).2. Simplify to \(2^{-4}\).3. Convert the negative exponent using the rule for negative exponents: \(\frac{1}{2^4}\).4. Calculate \(2^4\) which equals 16.5. Final answer is \(\frac{1}{16}\).This process highlights the importance of mastering both the rules of exponents and the method for handling negative exponents. By breaking down the steps:

• Identify the base and exponents clearly.
• Apply the correct exponent rule for the situation.
• Simplify gradually, focusing on using negative exponent rules correctly.
• Calculate any remaining powers with precision.

Practice these strategies regularly to develop confidence and ease with evaluating powers.