/*! 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 74 Combine like terms. $$ \frac... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 74

Combine like terms. $$ \frac{1}{10} y z^{4}-\frac{3}{4} y^{4} z+y z^{4}+\frac{3}{2} y^{4} z $$

Short Answer

Expert verified
Final simplified expression: \( \frac{11}{10} yz^4 + \frac{3}{4} y^4 z \).

Step by step solution

01

- Identify like terms

Identify the terms that have the same variables raised to the same powers. Here, the terms \(\frac{1}{10} yz^4\) and \(yz^4\) are like terms. Similarly, \(-\frac{3}{4} y^4 z\) and \(\frac{3}{2} y^4 z\) are like terms.
02

- Combine the like terms involving yz^4

Add the coefficients of \( \frac{1}{10} yz^4 \) and \( yz^4 \). Convert \( yz^4 \) into a fraction to match the denominators: \( 1 yz^4 = \frac{10}{10} yz^4 \). Now add: \(\frac{1}{10} yz^4 + \frac{10}{10} yz^4 = \frac{11}{10} yz^4\).
03

- Combine the like terms involving y^4z

Add the coefficients of \(- \frac{3}{4} y^4 z \) and \( \frac{3}{2} y^4 z \. Convert \( \frac{3}{2} \) to a fraction with the same denominator: \( \frac{3}{2} = \frac{6}{4} \). Now subtract: \- \frac{3}{4} y^4 z + \frac{6}{4} y^4 z = \frac{3}{4} y^4 z \).
04

- Write the final answer

Combine the results from the previous steps to get the final simplified expression: \( \frac{11}{10} yz^4 + \frac{3}{4} y^4 z \).

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and algebraic operations (addition, subtraction, multiplication, division). Examples include simple expressions like \(3x + 2\) and more complex ones like \(\frac{1}{10} yz^4 - \frac{3}{4} y^4 z + yz^4 + \frac{3}{2} y^4 z\).
Algebraic expressions are the foundation of algebra and they allow you to represent mathematical relationships and problems in a concise way.
You鈥檒l often need to simplify these expressions by combining like terms.
Polynomials
Polynomials are a type of algebraic expression where variables are raised to whole number exponents, and there are no division by variables. For instance, \(\frac{1}{10} yz^4\) and \(\frac{3}{2} y^4 z\) are terms in the polynomial given in your original exercise.* Polynomials can have one or more terms, like \(3x\) (monomial), \(3x + 2\) (binomial), or even more.
When working with polynomials, the goal is often to simplify them which frequently involves combining like terms.
Like Terms
Like terms are terms that have the same variables raised to the same powers. For instance, \(yz^4\) and \( \frac{1}{10} yz^4\) are like terms because they both include \( yz^4 \). Similarly, \( -\frac{3}{4} y^4 z \) and \( \frac{3}{2} y^4 z \) are like terms.
To combine like terms, you simply add or subtract their coefficients. This is a key step in simplifying algebraic expressions and polynomials.
Coefficients
Coefficients are the numerical parts of terms with variables. In the term \( \frac{1}{10} yz^4 \), \( \frac{1}{10} \) is the coefficient. In \( yz^4 \), the coefficient is understood to be \(1\).
When combining like terms, we add or subtract the coefficients while keeping the variables and their exponents unchanged. For example, adding \( \frac{1}{10} yz^4 \) and \( yz^4 \) involves adding \( \frac{1}{10} \) and \( 1 \), resulting in \( \frac{11}{10} yz^4 \).
Understanding how to work with coefficients is essential for simplifying expressions and solving algebraic equations.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. a. Determine the total expenditure for private health insurance premiums for the year \(2010 .\) b. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2010 .\) c. Evaluate the polynomial \(I+P\) found in Exercise \(57(\) a) for \(x=10\).

Determine the sign of the expression. Assume that \(a, b,\) and \(c\) are real numbers and \(a<0, b>0,\) and \(c<0\). $$\frac{a^{2} c}{b^{4}}$$

We know that \((a-b)^{2}=a^{2}-2 a b+b^{2}\). Derive a special product formula for \((a-b)^{3}\).

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ \left(2 c^{3} \sqrt{d}+3 d^{3} \sqrt{c}\right)^{2} $$

Perform the indicated operations and simplify. $$ [(c+d)-a][(c+d)+a] $$
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