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Chapter 8: Problem 4

Compute the exact value. $$ \left(\frac{2}{3}\right)^{1} $$

Short Answer

Expert verified
The exact value is \( \frac{2}{3} \).

Step by step solution

01

Power of One

When any number or fraction is raised to the power of 1, it remains unchanged. Thus, we have the expression: \( \left(\frac{2}{3}\right)^{1} = \frac{2}{3} \).
02

Final Result

Since the power does not change the value, the exact answer is simply the original fraction: \( \frac{2}{3} \).

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Key Concepts

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

Understanding Fraction Exponents
Fraction exponents might sound complicated at first, but they are simply a way of expressing powers and roots in a single notation. It鈥檚 useful to understand how they work to fully appreciate their role in algebra. Fraction exponents are used to signify both an exponent and a root. For example:
  • The expression \( a^{1/n} \) means the \( n \)-th root of \( a \).
  • Similarly, \( a^{m/n} \) means you first take the \( n \)-th root of \( a \), and then raise the result to the \( m \)-th power.
This combination allows for streamlined calculations without the need for separate root and power operations. In the given exercise, the exponent \( 1 \) is a fraction with the numerator \( 1 \) and an implicit denominator of \( 1 \), indicating the value remains unchanged.
Exploring Mathematical Expressions
Mathematical expressions are combinations of numbers, symbols, and operations that stand for a value or represent relationships among quantities. They are foundational in algebra and understanding them is crucial when solving equations. A mathematical expression can be as simple as a number, or more complex involving exponents, fractions, and variables.
  • In the given exercise, \( \left(\frac{2}{3}\right)^{1} \), the expression is very simple because it involves just a fraction and an exponent of one.
  • Understanding that the power of one doesn鈥檛 alter the base fraction simplifies how we view and interpret such expressions.
Recognizing simple expressions within more complex equations allows for easier solution strategies, especially when larger algebraic structures are involved. This basic knowledge helps students to efficiently tackle problems.
Core Algebra Concepts in Practice
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's about finding the unknowns and forming equations to express general relationships. Central to algebra is understanding how operations like addition, subtraction, multiplication, and division apply to variables, constants, and expressions.Let's explore some key algebra concepts related to this exercise:
  • Identity Properties: These are rules like the one that states any number raised to the power of one is itself. It鈥檚 exemplified perfectly in the example \( \left(\frac{2}{3}\right)^{1} = \frac{2}{3} \).
  • Fractional Operations: While the problem involves fractions, it's quite simple because of the power of one. However, understanding how fractions work and how they are manipulated in equations is fundamental.
These concepts form the building blocks for more advanced topics in algebra. Mastering them provides a solid foundation for tackling the challenges posed by more complex expressions and equations in higher levels of mathematics.

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Most popular questions from this chapter

Simplify the product, and write your answer in the form \(x^{r}\). $$ x^{-\frac{5}{4}} x^{\frac{5}{2}} $$

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Simplify the given radical expression. $$ \sqrt{121} $$

Compute the exact value. $$ \left(\frac{1}{3}\right)^{-3} $$

Compute the exact value. $$ 4^{-3} $$
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