/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Simplify the expression without ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 1

Simplify the expression without using a calculating utility. $$ \begin{array}{lll}{\text { (a) }-8^{2 / 3}} & {\text { (b) }(-8)^{2 / 3}} & {\text { (c) } 8^{-2 / 3}}\end{array} $$

Short Answer

Expert verified
(a) -4, (b) 4, (c) \(\frac{1}{4}\).

Step by step solution

01

Simplify Expression (a)

For expression (a) \(-8^{2/3}\), we interpret this as an exponent applied to 8, which simplifies to \(-1 \times 8^{2/3}\). Evaluate \(8^{2/3}\) by recognizing it uses a cube root and a square: 8 can be written as \(2^3\), so \(8^{1/3} = 2\). Thus, \(8^{2/3} = (2)^2 = 4\). Therefore, the expression is \(-1 \times 4 = -4\).
02

Simplify Expression (b)

For expression (b) \((-8)^{2/3}\), the negative sign is inside the exponentiation. First, recognize that \(-8 = (-2)^3\). Then \((-8)^{1/3} = (-2)\). Raise this to the power of 2 to get: \((-2)^2 = 4\). So the expression simplifies to 4.
03

Simplify Expression (c)

For expression (c) \(8^{-2/3}\), apply the negative exponent rule: \(a^{-n} = \frac{1}{a^n}\). Thus, \(8^{-2/3} = \frac{1}{8^{2/3}}\). From previous steps, we know \(8^{2/3} = 4\). So the expression simplifies to \(\frac{1}{4}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Negative Exponents
Negative exponents might seem confusing, but they're not too tricky once you get the hang of them. A negative exponent indicates that you should take the reciprocal of the base raised to the positive of that exponent. So, for example, with the expression \(8^{-2/3}\), the negative exponent means we will flip the base.
  • With any expression \(a^{-n}\), know it becomes \(\frac{1}{a^n}\).
  • In practice, this means the operation involves dividing 1 by the base raised to the positive exponent.
Returning to the example \(8^{-2/3}\): we rewrite it as \(\frac{1}{8^{2/3}}\). Then you simplify \(8^{2/3}\) as needed, which, in our exercise, has been calculated to be 4. Therefore, the expression can be simplified to \(\frac{1}{4}\). Breaking down negative exponents in this way demystifies their purpose and usage.
Fractional Exponents
Fractional exponents can seem complex because they involve roots and powers, but when understood correctly, they're straightforward. A fractional exponent like \(8^{2/3}\) includes both a root and a power:
  • The denominator gives you the root. For instance, \(8^{1/3}\) means "the cube root of 8."
  • The numerator gives you the power. So in \(8^{2/3}\), we raise the result of \(8^{1/3}\) to the second power.
Let's break it down further with the example: we recognize 8 as \(2^3\), meaning \(8^{1/3} = 2\). Then, the original fractional exponent \(8^{2/3}\) becomes \((2)^2 = 4\). Understanding the dual nature of fractional exponents, involving both roots and powers, is key to simplifying such expressions.
Simplifying Expressions
Simplifying expressions can involve many steps, especially when dealing with exponents, and it's all about breaking down the parts into manageable pieces. When approached methodically, it becomes much simpler:
  • Identify the type of exponent: Is it positive, negative, or fractional?
  • Apply the appropriate rules: Use laws of exponents such as \(a^{-n} = \frac{1}{a^n}\) for negative exponents.
  • Break down complex parts: For fractional exponents, handle the root first before applying any further powers.
Take expression \(-8^{2/3}\) as a simplification example: it's understood as \(-1 \times 8^{2/3}\), differing from \((-8)^{2/3}\), showcasing the importance of order and understanding parenthesis influence. By reducing such expressions step-by-step, even the most daunting problems become much more approachable. Following a consistent method helps manage expressions efficiently and assures accurate solutions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exact value of the expression without using a calculating utility. $$ \begin{array}{ll}{\text { (a) } \log _{2} 16} & {\text { (b) } \log _{2}\left(\frac{1}{32}\right)} \\ {\text { (c) } \log _{4} 4} & {\text { (d) } \log _{9} 3}\end{array} $$

Sketch the graph of the equation by making appropriate transformations to the graph of a basic power function. If you have a graphing utility, use it to check your work. $$ \begin{array}{ll}{\text { (a) } y=1-\sqrt{x+2}} & {\text { (b) } y=1-\sqrt[3]{x+2}} \\ {\text { (c) } y=\frac{5}{(1-x)^{3}}} & {\text { (d) } y=\frac{2}{(4+x)^{4}}}\end{array} $$

A variable \(y\) is said to be inversely proportional to the square of a variable \(x\) if \(y\) is related to \(x\) by an equation of the form \(y=k / x^{2},\) where \(k\) is a nonzero constant, called the constant of proportionality. This terminology is used in these exercises. According to Coulomb's \(\operatorname{law},\) the force \(F\) of attraction between positive and negative point charges is inversely proportional to the square of the distance \(x\) between them. (a) Assuming that the force of attraction between two point charges is 0.0005 newton when the distance between them is 0.3 meter, find the constant of proportionality (with proper units). (b) Find the force of attraction between the point charges when they are 3 meters apart. (c) Make a graph of force versus distance for the two charges. (d) What happens to the force as the particles get closer and closer together? What happens as they get farther and farther apart?

Find a formula for \(f^{-1}(x)\) $$ f(x)=3 x^{3}-5 $$

Solve for \(x\) without using a calculating utility. Use the natural logarithm anywhere that logarithms are needed. $$ 3 e^{-2 x}=5 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.