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Chapter 2: Problem 58

Simplify the given expression as completely as possible. $$(-4 x)\left(6 x^{5}\right)$$

Short Answer

Expert verified
-24\(x^6\)

Step by step solution

01

Distribute the Coefficients

Multiply the numerical coefficients together. The coefficients are -4 and 6. Thus, \(-4 \times 6 = -24\).
02

Multiply the Variables

Multiply the variables by adding their exponents. The expression has variables \(x\) and \(x^5\). This gives us \(x^{1+5} = x^6\).
03

Combine the Results

Combine the results from Step 1 and Step 2 to get the simplified expression. The final expression is \(-24x^6\).

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Key Concepts

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

Multiplying Coefficients
When simplifying algebraic expressions, multiplying coefficients is often the first step. Coefficients are the numerical parts of the terms. In the given exercise, we have two coefficients: \(-4 \) and \( 6 \). To multiply them, simply perform the arithmetic multiplication: \(-4 \) multiplied by \( 6 \) equals \(-24 \). This is a straightforward process. Here are the steps in detail:
  • Identify the coefficients in the expression.
  • Multiply the coefficients using basic multiplication rules.
  • Combine the product with the remaining parts of the expression.
By completing this, we get the new coefficient of the simplified term, which is \(-24 \).
Remember, always multiply the numerical parts first before handling any variables.
Adding Exponents
The next step is to multiply the variables. In algebra, when multiplying variables that have exponents, you need to add the exponents together. Let's take a closer look:
In the given exercise, the variables are \(x\) and \(x^5\). When multiplying these, add their exponents. The exponent for \(x\) is \(1\) (since \(x = x^1\)). So, you have: \(x^1 \times x^5 = x^{1+5} = x^6\).
Let's break it down:
  • Identify the exponents of the variables involved.
  • Add the exponents together to get the new exponent.
  • Write the variable with the new exponent.
Therefore, we get \(x^6\). This step ensures we correctly handle the powers of the variables while simplifying the expression.
Combining Like Terms
Finally, we need to combine the results from the previous steps. In algebraic expressions, like terms are terms that have the same variables raised to the same power.
Combining them involves:
  • Making sure the variables and their exponents match up.
  • Combining their coefficients.
In the given exercise, after multiplying the coefficients and adding the exponents, the simplified expression is: \(-24x^6\). This is already in its simplest form since there are no other terms to combine with.

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Most popular questions from this chapter

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