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Chapter 1: Problem 66

Simplify each expression. \(5 y^{3}+6 y^{3}-3 y^{2}-4 y^{2}\)

Short Answer

Expert verified
11y^3 - 7y^2

Step by step solution

01

Combine like terms for the cubic terms

Identify and combine terms with the same power of 3. These terms are: 5y^3 and 6y^3. Add them together: 5y^3 + 6y^3 = 11y^3
02

Combine like terms for the square terms

Identify and combine terms with the same power of 2. These terms are: -3y^2 and -4y^2. Add them together: -3y^2 - 4y^2 = -7y^2
03

Combine the results from steps 1 and 2

Put together the simplified terms from the previous steps to form the final simplified expression: 11y^3 - 7y^2

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Key Concepts

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

Combining Like Terms
When simplifying expressions, one of the fundamental steps is combining like terms. Like terms are terms that have the same variable raised to the same power. In our example, we have the terms 5y^{3} and 6y^{3}, both of which are cubic terms because they have the variable y raised to the power of three. Similarly, -3y^{2} and -4y^{2} are like terms because they have the variable y raised to the power of two.

To combine like terms, you simply add or subtract the coefficients (the numerical parts) while keeping the variable part unchanged. Here is how you do it:
  • Identify the like terms in the expression.
  • Add or subtract their coefficients.
  • Keep the variable and its exponent the same.
For example, combining 5y^{3} and 6y^{3} involves adding 5 and 6, resulting in 11y^{3}.
Cubic Terms
Cubic terms are terms in an expression where the variable is raised to the power of three. These terms represent the volume of a cube whose sides are the length of the variable. In the original expression, we have two cubic terms: 5y^{3} and 6y^{3}.

To simplify expressions with cubic terms, first, identify all the cubic terms. Then, combine them by adding or subtracting their coefficients. For instance:
  • Identify cubic terms: 5y^{3} and 6y^{3}.
  • Combine them: 5y^{3} + 6y^{3} = 11y^{3}.
The result is 11y^{3}, a simplified cubic term.
Square Terms
Square terms are terms in an expression where the variable is raised to the power of two. These represent the area of a square whose sides are the length of the variable. In our example, the square terms are -3y^{2} and -4y^{2}.

When simplifying square terms, follow the same process as with other like terms: find and combine them. Let's do it step by step:
  • Identify the square terms: -3y^{2} and -4y^{2}.
  • Combine them: -3y^{2} - 4y^{2} = -7y^{2}.
The final result here is -7y^{2}, which is a simplified square term.

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