Problem 21 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$\frac{2^{3}}{2^{7}}$$

Step-by-Step Solution

Answer

The value of the given exponential expression is 1/16

Step 1: Understand the Basic Rule

Understand the basic rule of exponents that states \((a^{m}) / (a^{n}) = a^{(m-n)}\). Where \(a\) is the base, \(m\) and \(n\) are the exponents.

Step 2: Apply the Rule

Apply this rule to the given expression \((2^{3}) /(2^{7})\). This can be written as \(2^{(3-7)}\), because the base is same (2) and we are simply subtracting the power of denominator from the power of numerator.

Step 3: Evaluate

Evaluate the exponent to get the final answer \(2^{-4}\). But since it is a negative exponent, the rule of exponents states that we should reciprocate the base. Thus, the answer would be \(1 / (2^{4}) = 1/16\).

Key Concepts

Laws of Exponents

The laws of exponents are foundational rules that help simplify expressions involving powers of the same base. These rules apply when multiplying, dividing, or raising numbers to powers. A basic law to remember is:

• Division Law: When you divide two exponents with the same base, subtract the exponent in the denominator from the exponent in the numerator. This is expressed as:
  • \(a^m / a^n = a^{m-n}\). Where \(a\) is the base, and \(m\) and \(n\) are the exponents.

This rule helps reduce the complexity of expressions. For example, given \((2^3)/(2^7)\):

• You recognize that the base is the same: 2.
• By applying the division law, you subtract the exponents: \(3 - 7 = -4.\)

This simplification shows the power of consistent rules in math to make complex calculations easier.

Negative Exponents

Negative exponents represent how many times you need to divide 1 by the base. They offer an elegant way to express the reciprocal of a number. For instance, \(a^{-n}\) can be rewritten as \(1/a^n\).

• When you encounter a negative exponent, flip the base to the reciprocal to make it positive.
• For \(2^{-4}\), the negative sign instructs us to take the reciprocal: \(1/2^4\).

This approach is particularly useful in scientific notation and algebra, making calculations more straightforward. After determining the positive expression, calculate as usual: So for \(2^{-4}\), which becomes \(1/16,\) you're simply finding

• \(2 \times 2 \times 2 \times 2 = 16,\)

then taking the reciprocal, giving you \(1/16\).

Fractional Exponents

Fractional exponents can be a bit puzzling at first, but they unlock the magic of expressing roots as exponents. The notation \(a^{m/n}\) provides a connection between powers and roots. Here’s how it breaks down:

• The denominator \(n\) indicates the type of root, and
• The numerator \(m\) represents the power you raise the base to.

To illustrate, consider \(a^{1/2}.\) This is essentially the square root of \(a\). The beautiful versatility of exponential expressions through fractional exponents allows mathematicians to seamlessly transition between powers and roots.