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Chapter 1: Problem 23

Evaluate each expression without using a calculator. $$ (-8)^{2 / 3} $$

Short Answer

Expert verified
The value of \((-8)^{2/3}\) is 4.

Step by step solution

01

Understand the Expression

The expression \((-8)^{2/3}\) means that we need to raise -8 to the power of \(\frac{2}{3}\). This is a fractional exponent, which can be broken down into two parts: a cube root and a square.
02

Cube Root of -8

Find the cube root of -8. The cube root of a number \(x\) is another number \(y\) such that \(y^3 = x\). Since \((-2)^3 = -8\), the cube root of -8 is -2. Therefore, \((-8)^{1/3} = -2\).
03

Square the Cube Root Result

Now take the cube root result, which is -2, and raise it to the power of 2. So we need to calculate \((-2)^2\). Squaring -2 means multiplying -2 by itself: \[ (-2) \times (-2) = 4 \]
04

Combine the Results

The result of the expression \((-8)^{2/3}\) is 4 because we've calculated the cube root of -8 to be -2 and squaring it gives 4.

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Key Concepts

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

Cube Roots
The cube root of a number is one of the essential algebraic operations, especially when dealing with fractional exponents. To find the cube root of a number like -8, we're looking for a number that, when multiplied by itself three times, gives us -8. In mathematical terms, if \( y^3 = x \), then \( y \) is the cube root of \( x \).

When dealing with negative numbers, it's important to remember that the cube root of a negative number will also be negative. This is because a negative number raised to an odd power (like 3) still results in a negative number. Therefore, the cube root of -8 is -2 because \((-2)^3 = -8\).
  • Key Point: The cube root of -8 is -2.
  • Understanding: We find -2 because it satisfies the equation \( (-2)^3 = -8 \).
  • Conceptual Tip: Remember that only one cube root exists for any real number, unlike square roots, which can yield both positive and negative solutions.
Exponentiation
Exponentiation refers to raising a number to a power, essential in algebra for simplifying and solving expressions. A fractional exponent, like \( \frac{2}{3} \), implies a sequence of operations: first finding the root and then raising to a power, or vice versa. For example, with \((-8)^{2/3}\), you first find the cube root, then square the result.

Consider the fractional exponent formula, where \( x^{m/n} \), means taking the \( n \)-th root of \( x \) first, and then raising the result to the \( m \)-th power.
  • Example: For \((-8)^{2/3}\), the cube root of -8 is -2, and squaring this results in 4.
  • Breaking It Down: First find the cube root \((-8)^{1/3} = -2\), then square it \((-2)^2 = 4\).
Exponentiation simplifies complex expressions and solves equations quickly, making it indispensable in algebra.
Algebraic Operations
Algebraic operations, including exponentiation, addition, subtraction, and roots, are foundational to solving and simplifying equations. Understanding these can significantly improve problem-solving skills.

The concept of fractional exponents mixes roots and powers, as seen in the example \((-8)^{2/3}\). It necessitates combining operations: first finding a root and then applying exponentiation.
  • Step-by-Step: Evaluate by using algebraic operations in sequence: cube root first, then exponentiation.
  • Importance: These operations allow us to manipulate equations and expressions to find solutions effortlessly.
  • Tip: When evaluating fractional exponents, clearly separate the root and the exponent parts for clarity.
Algebraic operations are essential in daily calculations and higher mathematics, forming a critical skill in math education.

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Most popular questions from this chapter

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ f(x)=5 x^{2} $$

For each function, find and simplify \(f(x+h)\). $$ f(x)=3 x^{2}-5 x+2 $$

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=x^{2}-x ; g(x)=\frac{x^{3}-1}{x^{3}+1} $$

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=x^{2}+x^{1 / 2} $$

ECONOMICS: Income Tax The following function expresses an income tax that is \(15 \%\) for incomes below $$\$ 6000$$, and otherwise is $$\$ 900$$ plus \(40 \%\) of income in excess of $$\$ 6000$$. \(f(x)=\left\\{\begin{array}{ll}0.15 x & \text { if } 0 \leq x<6000 \\\ 900+0.40(x-6000) & \text { if } x \geq 6000\end{array}\right.\) a. Calculate the tax on an income of $$\$ 3000$$. b. Calculate the tax on an income of $$\$ 6000$$. c. Calculate the tax on an income of $$\$ 10,000$$. d. Graph the function.
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