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Chapter 10: Problem 16

In the following exercises, add or subtract the monomials. $$4 y^{3}+6 y^{3}$$

Short Answer

Expert verified
10y^3

Step by step solution

01

- Identify the Monomials

Look at the given terms: \(4y^3\) and \(6y^3\). These are the monomials you need to add.
02

- Check for Like Terms

Ensure that the terms are like terms. Like terms have the same variable raised to the same power. Here, both terms have the variable \(y\) raised to the power of 3, so they are like terms.
03

- Add the Coefficients

Add the coefficients of the like terms. The coefficients are 4 and 6. So, calculate: \(4 + 6 = 10\)
04

- Write the Result with the Variable

Attach the common variable part \(y^3\) to the sum of the coefficients calculated in Step 3. The result is: \(10y^3\)

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Key Concepts

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

Monomials
To understand adding and subtracting monomials, you first need to know what a monomial is. A monomial is a single term algebraic expression that consists of a coefficient and variables that are raised to non-negative integer powers. For example, in the term [4y^3], 4 is the coefficient, and [y^3] is the variable part.

A monomial can be as simple as a constant number like 5, or it can include variables and exponents, like [3x^2] or [-7ab^3].
Monomials are the building blocks in algebra that you will combine in various ways.
Like Terms
When working with monomials, it is crucial to identify like terms.

Like terms are terms that have exactly the same variable(s) raised to the same power(s). In the given exercise, [4y^3] and [6y^3] are like terms because both have the variable [y] raised to the power of 3.

Here's how to spot like terms:
  • Look for identical variables
  • Check that these variables are raised to the same exponent
Only like terms can be added or subtracted. This is because adding or subtracting unlike terms (terms with different variables or exponents) doesn't conform to the rules of algebraic operations.
Coefficients
Coefficients are the numerical part of a term.

In the equation [4y^3 + 6y^3], the coefficients are 4 and 6. To add or subtract monomials, you focus primarily on combining the coefficients while keeping the variable part unchanged.

For example, when you add [4y^3] and [6y^3], you add the coefficients (4 + 6) to get 10, and then attach the common variable part [y^3] to the result. This gives you [10y^3].

So, the process involves:
  • Identifying the coefficients
  • Ensuring the terms are like terms
  • Adding or subtracting the coefficients
  • Keeping the common variable part unchanged
Understanding coefficients is key to mastering the addition and subtraction of monomials.

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