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Chapter 5: Problem 74

Perform the indicated operations. $$\left(w^{2}-a\right)\left(t^{2}+3\right)$$

Short Answer

Expert verified
w^{2}t^{2} + 3w^{2} - at^{2} - 3a

Step by step solution

01

Distribute each term in the first parenthesis

To perform the indicated operations, start by distributing each term in the first parenthesis to each term in the second parenthesis. Multiply each term of \( w^{2}-a \) by each term of \( t^{2}+3 \).
02

Multiply the first term in the first parenthesis

First, multiply \( w^{2} \) by each term in the second parenthesis: \[ w^{2} \times t^{2} + w^{2} \times 3 = w^{2}t^{2} + 3w^{2} \].
03

Multiply the second term in the first parenthesis

Next, multiply \( -a \) by each term in the second parenthesis: \[-a \times t^{2} + -a \times 3 = -at^{2} - 3a \].
04

Combine all terms

Finally, combine all the terms obtained from the previous multiplications: \[ w^{2}t^{2} + 3w^{2} - at^{2} - 3a \].

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. In symbols, this is written as: a(b + c) = ab + ac In our exercise, we have \(w^{2} - a\) and \(t^{2} + 3\). To multiply these, we distribute each term in the first set of parentheses to each term in the second set of parentheses. Here鈥檚 what happens:
  • First, we take \(w^{2}\) and multiply it by both \(t^{2}\) and \(3\).
  • Then, we take \(-a\) and multiply it by both \(t^{2}\) and \(3\).
By breaking down the problem into these smaller steps, we make it more manageable.
combining like terms
After distributing, we end up with terms that may need to be combined. Combining like terms simplifies our expression and can make it easier to work with. Like terms are terms that have the same variables raised to the same powers. For example, in the expression: \( w^2t^2 + 3w^2 - at^2 - 3a \)
  • \(w^2t^2\) and \(3w^2\) are not like terms because they have different variables and powers.
  • \(3w^2\) and \(-at^2\) are also not like terms.
  • Finally, the terms \( - 3a \) has a different variable and power compared to any of the others.
Since none of these terms are like terms, we cannot combine any further. The expression is as simplified as it can be.
algebraic expressions
An algebraic expression is a combination of numbers, variables, and operations. In our exercise, \( (w^2 - a)(t^2 + 3) \) is an algebraic expression.This expression involves both multiplication and addition. To solve it, we follow a system of operations.
  • First, we use the distributive property to expand the expression.
  • Then, we combine any like terms (if possible).
Understanding how to manipulate algebraic expressions is crucial in algebra. It helps in solving equations, simplifying results, and understanding mathematical relationships. Always take note of the variables and perform the operations step by step.

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