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Chapter 0: Problem 21

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

Short Answer

Expert verified
The value of the given exponential expression is 1/16

Step by step solution

01

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.
02

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.
03

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 \(1 / (2^{4}) = 1/16\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

Find the exact value of \(\sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}\) without the use of a calculator.

How do you know if a number is written in scientific notation?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Special-product formulas have patterns that make their multiplications quicker than using the FOIL method.

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