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Chapter 5: Problem 12

Multiply. $$ \left(-x^{3}\right)\left(-x^{5}\right) $$

Short Answer

Expert verified
The answer is \( x^8 \) .

Step by step solution

01

- Analyze the Problem

The given problem is to multiply two expressions: \( -x^3 \) and \( -x^5 \).
02

- Multiply the Coefficients

First, multiply the coefficients of the expressions. Here, we have \( -1 \) and \( -1 \). \(-1 \times -1 = 1\).
03

- Apply the Power Rule for Exponents

Next, use the power rule for exponents that states \( x^a \times x^b = x^{a+b} \). In this problem, multiply \( x^3 \) and \( x^5 \) to get \( x^{3+5} = x^8 \).
04

- Combine Results

Combine the results from Step 2 and Step 3. The final expression is \( 1 \times x^8 = x^8 \).

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Key Concepts

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

coefficients
In mathematics, a coefficient is a number or constant placed in front of and multiplying a variable in an algebraic expression.
For example, in the expression \(3x^2\), the number \(3\) is the coefficient and \(x^2\) is the term or variable.
Coefficients can be positive or negative numbers and they play a crucial role in multiplication of polynomials.
In our exercise, we have the terms \(-x^3\) and \(-x^5\). Here, the coefficients are \(-1\) for both terms.
When multiplying these coefficients \(-1 \times -1\), we get a positive result, \(1\).
This step simplifies our problem and sets the foundation for combining the variable parts of the expression.
power rule for exponents
The power rule for exponents is an essential rule when dealing with polynomials and variable expressions.
According to this rule, when you multiply two expressions with the same base, you add their exponents: \(x^a \times x^b = x^{a+b}\).
In our exercise, we have two expressions: \(-x^3\) and \(-x^5\). After handling the coefficients, we focus on the variables \(x^3\) and \(x^5\).
Applying the power rule, we add the exponents: \3 + 5 = 8\.
Therefore, multiplying \(x^3\) by \(x^5\) results in \(x^8\). This step allows us to simplify the expression further and prepare for combining all parts of our solution.
combining like terms
Combining like terms is a vital step in simplifying algebraic expressions.
Like terms are terms that have the same variable raised to the same power.
In our exercise, after multiplying the coefficients and applying the power rule, we are left with \(1 \times x^8\).
Since there are no other terms to combine, our final result is simply \(x^8\).
The process of combining like terms helps in simplifying complex expressions, making it easier to understand and solve the equations.
This principle applies not only to multiplication but also to addition, subtraction, and other algebraic operations.

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