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Chapter 3: Problem 415

In the following exercises, evaluate each expression. $$ (x+y)^{3} \text { when } x=-4, y=1 $$

Short Answer

Expert verified
-27

Step by step solution

01

Substitute the Values

Substitute the given values of \(x\) and \(y\) into the expression. Given \(x = -4\) and \(y = 1\), the expression \((x + y)^3\) becomes \((-4 + 1)^3\).
02

Simplify Inside the Parentheses

First, simplify inside the parentheses: \((-4 + 1) = -3\). So the expression now is \((-3)^3\).
03

Compute the Cube

Evaluate the cube of \(-3\): \((-3)^3 = -3 \times -3 \times -3 = -27\).

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Key Concepts

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

Substitution in Expressions
Substituting values into an expression is a key step in evaluating mathematical problems. It's like swapping out variables for actual numbers.
Let's break it down further:
First, identify the variables in the expression. They are the letters, like \(x\) and \(y\), that will be replaced by specific values.
In our problem, we have \(x = -4\) and \(y = 1\).
Next, plug these values into the expression. In this case, \((x + y)^3\) becomes \((-4 + 1)^3\). By directly substituting the values, we transform a complex algebraic expression into a simpler arithmetic one.
Simplifying Expressions
Simplifying expressions involves making them easier to understand and solve.
For our exercise, we start with \((-4 + 1)^3\).
First, perform the operation inside the parentheses. Adding -4 and 1 gives -3, so we simplify to \((-3)^3\).
Breaking down challenges piece by piece makes them less overwhelming.
Tips for simplifying:
  • Always deal with parentheses first.
  • Combine like terms, if any.
  • Follow the order of operations (PEMDAS/BODMAS).
By simplifying expressions step by step, we maintain clarity and avoid errors.
Cubing Negative Numbers
Cubing negative numbers can be tricky, but understanding the process helps.
To cube a number means multiplying it by itself twice.
For example, let's cube \(-3\).
\((-3)^3 = -3 \times -3 \times -3\).
First, multiply -3 by -3, which equals 9 (remember, a negative times a negative is positive).
Then, multiply this 9 by -3 again. Now, a positive times a negative gives a negative, so the result is -27.
Key points to remember:
  • Cubing a negative number results in a negative.
  • Cubing a positive number results in a positive.
  • Even exponents of negatives make positives; odd exponents keep them negative.
Understanding these basic principles makes cubing negative numbers simple.

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