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Chapter 5: Problem 9

Multiply. $$ \left(-x^{2}\right)(-x) $$

Short Answer

Expert verified
The answer is \(x^{3}\).

Step by step solution

01

Understand the Problem

The task is to multiply two algebraic expressions: \(-x^{2}\) and \(-x\).
02

Apply the Sign Rule

When multiplying two negative numbers, the result is positive. Therefore, \(-x^{2}\) multiplied by \(-x\) will result in a positive expression.
03

Multiply the Coefficients

Since both terms have an implied coefficient of 1, multiply these coefficients together: \(-1 \times -1 = 1\).
04

Multiply the Variables

Combine the exponents of the variable \(x\). Recall the rule \(x^{a} \times x^{b} = x^{(a + b)}\). Hence the terms \(x^{2}\) and \(x\) combine to \(x^{(2 + 1)} = x^{3}\).
05

Combine the Results

Combine the results from Steps 3 and 4. The multiplication is: \(1 \times x^{3} = x^{3}\).

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Key Concepts

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

Sign Rules
When you multiply algebraic expressions, paying attention to sign rules is crucial. If you're dealing with negative numbers, remember:
  • Negative multiplied by negative gives a positive result.
  • Positive multiplied by negative gives a negative result.
  • Positive multiplied by positive gives a positive result.

So, if you have \(-x^2(-x)\), first address the signs. Both \(-x^2\) and \(-x\) are negative. Therefore, their product is positive. This simplifies our work early on and helps avoid errors.
Multiplying Coefficients
Next, let's focus on coefficients. Coefficients are the numbers in front of the variables. In \(-x^2\) and \(-x\), the coefficients are -1 (because there's no visible number, it's implied to be 1).

  • Multiply \(-1\) by \(-1\): \(-1 \times -1 = 1\).

This step is often straightforward but easy to miss, so always check your coefficients.
Combining Exponents
Now, let's combine the variables and their exponents. Consider the basic rule: \(x^a \times x^b = x^{a+b}\). This means when you multiply similar bases (the variable \x\ here), you add the exponents together.

  • For \(-x^2\) and \(-x\), first disregard the coefficients (we already handled those).
  • Combine \(x^2\) and \(x\): \(2 + 1 = 3\).

So, \(x^2 \times x = x^3\). Combining this with our coefficient from the previous step, we get \(1 \times x^3 = x^3\).

This gives us the final simplified product of the original expression.

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Most popular questions from this chapter

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graph to check each prediction.

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x^{2}\\\ &g(x)=7 x-4 \end{aligned} $$

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=64 x^{3}-8, g(x)=4 x-2 $$

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{14} $$

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=6 x^{2}-11 x-10, g(x)=3 x+2 $$
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