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Chapter 0: Problem 21

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

Short Answer

Expert verified
The evaluated expression equals \(1 / 16\).

Step by step solution

01

- Identify the bases and exponents

In the expression, the base is 2 and the exponents are 3 and 7 for the numerator and denominator respectively, i.e., \(2^{3}\) and \(2^{7}\).
02

- Apply the rule of exponents

The rule of exponents states that for any non-zero number a and any integers m and n, \(a^{m} / a^{n} = a^{m-n}\). Hence, divide the bases by subtracting the exponents, i.e., \(2^{3-7}\).
03

- Simplify the result

The value of the expression \(2^{3-7}\) equals \(2^{-4}\). The negative exponent means we change the fraction to its reciprocal, hence, the answer is \(1 / 2^{4}\).
04

- Evaluate the power

Compute \(2^{4}\), which equals 16.

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Key Concepts

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

Exponents
In mathematics, an exponent refers to the number that is placed above and to the right of a base number. This concept indicates how many times the base is multiplied by itself. For instance, in the expression \(2^3\), the base is 2, and the exponent is 3, meaning 2 is multiplied by itself three times: \(2 \times 2 \times 2\). Exponents are a concise way to express repeated multiplication, making calculations easier and more efficient. They play a vital role in simplifying large numbers and solving various algebraic problems, as they reduce complex expressions to more manageable terms.
Fractional Exponents
Fractional exponents are a type of exponent that represents both a power and a root. When you see an expression like \(a^{m/n}\), this means you're looking at the \(n\)-th root of \(a\), raised to the \(m\)-th power. For example, \(8^{1/3}\) denotes the cube root of 8.
  • The numerator of the fractional exponent (\(m\)) indicates the power.
  • The denominator of the fractional exponent (\(n\)) indicates the root.
Understanding fractional exponents allows us to solve equations that contain roots more efficiently, as they transform roots into multipliable expressions. Learning to switch between radical notation and fractional exponents can greatly streamline problem-solving processes.
Rules of Exponents
The rules of exponents are essential for simplifying expressions and solving equations that contain exponents. One basic rule, applied in the given problem, is the quotient of powers rule, which states: if you divide like bases, you subtract the exponents. That is, for any non-zero number \(a\) and integers \(m\) and \(n\), \(a^m / a^n = a^{m-n}\).
Other important rules include:
  • Product of powers rule: \(a^m \times a^n = a^{m+n}\).
  • Power of a power rule: \((a^m)^n = a^{mn}\).
  • Zero exponent rule: \(a^0 = 1\) for \(a eq 0\).
  • Negative exponent rule: \(a^{-n} = 1/a^n\).
Mastering these rules helps in dealing with complex algebraic expressions efficiently and accurately. They offer a structured approach to breaking down and simplifying calculations effectively.

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Most popular questions from this chapter

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

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Will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200,\) write a simplified algebraic expression that models the rectangle's perimeter.
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