/*! 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 16 Decide whether each expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 16

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ \frac{2^{0}}{0^{2}+2^{0}} $$

Short Answer

Expert verified
The expression simplifies to 1.

Step by step solution

01

- Simplify the numerator

Start by simplifying the numerator of the given expression. The numerator is given by the expression: \[ 2^0 \]Since any non-zero number raised to the power of 0 is equal to 1, we have:\[ 2^0 = 1 \]
02

- Simplify the denominator

Simplify the denominator, which is given by the expression: \[ 0^2 + 2^0 \]First, calculate both terms inside the denominator:\[ 0^2 = 0 \]\[ 2^0 = 1 \]Adding these results together gives:\[ 0 + 1 = 1 \]
03

- Divide the simplified numerator by the simplified denominator

Now, substitute the simplified numerator and denominator back into the expression to get:\[ \frac{2^0}{0^2 + 2^0} = \frac{1}{1} \]Since dividing 1 by 1 equals 1, the expression simplifies to:\[ \frac{1}{1} = 1 \]

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.

Exponentiation Rules
Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent. Exponentiation rules are crucial for simplifying algebraic expressions.
Here are some key exponentiation rules to keep in mind:
  • Any non-zero number raised to the power of zero equals 1. For example: \(\text{a}^0 = 1\), where \(\text{a} eq 0\).
  • The power of zero itself is always zero unless it is raised to zero, which is one of the few exceptions, like \(\text{0}^0 = 1\).
  • Multiply two numbers with the same base by adding their exponents: \[a^m \times a^n = a^{m+n}\]
  • Divide two numbers with the same base by subtracting their exponents: \[a^m / a^n = a^{m-n}\]
  • Raising a power to another power means multiplying the exponents: \((a^m)^n = a^{m \times n}\)
Understanding these rules will help you simplify complex algebraic expressions.
Basic Arithmetic Operations
Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are foundational in mathematics and are frequently used when simplifying expressions.
Let's go over them:
  • Addition (\text{+}): Combining two or more numbers to get their total or sum. \(3 + 2 = 5\).
  • Subtraction (\text{-}): Taking one number away from another. \(5 - 2 = 3\).
  • Multiplication (\text{x}): Adding a number to itself a certain number of times. \(4 \times 3 = 12\).
  • Division (\text{梅}): Distributing a number into a specific number of parts. \(12 梅 4 = 3\).
When simplifying an algebraic expression, it's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that the expression is simplified correctly. For example:
  • Simplify exponents first: \(2^0 = 1\).
  • Then, perform operations inside parentheses: \(0^2 + 2^0 = 0 + 1 = 1\).
  • Finally, perform division or multiplication: \(2^0 / (0^2 + 2^0) -> 1 / 1 = 1\).
Fraction Simplification
Simplifying fractions involves reducing them to their simplest form. A fraction consists of a numerator (top number) and a denominator (bottom number). To simplify a fraction, follow these steps:
  • Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCD.
For example, to simplify \(\frac{8}{12}\):
  • First, find their GCD, which is 4.
  • Divide both by 4: \( \frac{8 梅 4}{12 梅 4} = \frac{2}{3} \).
However, in the given expression \( \frac{2^0}{0^2 + 2^0} \), there's no need to find a common divisor because both the numerator and the denominator are 1 after simplification.
Therefore, \( \frac{1}{1} = 1 \). This demonstrates the process of simplifying fractions and also how basic understanding of exponentiation and arithmetic rules helps.

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 each product. $$ \left(x^{2}-2\right)\left(3 x^{2}+x+4\right) $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 2 x^{2}-3 x+10 $$

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{0.000000081(0.000036)}{0.00000048} $$

A polynomial in the variable \(x\) has degree 6 and is divided by a monomial in the variable \(x\) having degree \(4 .\) What is the degree of the quotient?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.