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Chapter 3: Problem 18

Multiply. $$-6 k^{5}(2 k)\left(-4 k^{2}\right)$$

Short Answer

Expert verified
Question: Multiply the following three terms: $$-6 k^{5}, (2k), and (-4 k^{2})$$. Answer: $$48k^{8}$$.

Step by step solution

01

Multiply coefficients

First, multiply the coefficients of the given terms: $$-6 \times 2 \times (-4) = -12 \times (-4) = 48$$
02

Add exponents of the variable \(k\)

Now, we need to add the exponents of the variable \(k\) in the given terms: \(k^{5} \times k \times k^{2}\). Remember that when multiplying terms with the same variable, we should add their exponents: $$k^{(5+1+2)} = k^{8}$$.
03

Combine the results

Finally, combine the results from Step 1 and Step 2 together: $$48k^{8}$$. This is the simplified form of the expression given by the exercise.

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Key Concepts

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

Coefficient Multiplication
Polynomial multiplication often involves multiplying coefficients, which are the constant numbers that accompany variables in terms. In the original exercise, we start by identifying and multiplying these coefficients together. Here, the coefficients given are
  • -6,
  • 2,
  • and -4.
The first step is to multiply these coefficients. Begin by multiplying -6 and 2, which equals -12. Next, multiply -12 by -4. Remember, when multiplying two negative numbers, the result is positive.

This gives us the final coefficient of 48. Multiplying coefficients is straightforward as it typically involves basic arithmetic. Make sure to keep track of the sign changes as they significantly affect the final value.
Exponent Rules
Exponent rules simplify the process of multiplying terms that share the same base, such as a variable. The key rule to remember is: when multiplying terms with the same base, you add their exponents.

For the terms in the exercise, we're working with the base variable k:
  • First term: k^{5},
  • Second term: k (which is k^{1}),
  • Third term: k^{2}.
Simply add the exponents together: 5 (from k^{5}) + 1 (from k) + 2 (from k^{2}) = 8.

This gives us the simplified form k^8 for the variable component of the expression. Remember, this rule only applies when the base remains the same across all terms being multiplied.
Variable Multiplication
Multiplying variables in polynomial multiplication follows the same principle as coefficient multiplication, but involves tracking the power of each variable. Each variable carries an exponent that dictates how many times it is multiplied by itself.

In this exercise, since all terms share the common variable k, we utilize exponent rules to determine the combined power (k^{8}).
  • Treat the variables with exponents like the coefficients, focusing on the addition of exponents when multiplying,
  • A new, singular term accounts for this combined exponent.
Maintaining this step-by-step approach ensures clarity and accuracy during calculation.
Simplifying Expressions
Once the coefficients and variable exponents have been handled separately, the next step is to combine them into a single simplified expression. This involves pairing the results from both coefficient multiplication and exponent addition.

In the original problem, the simplified result after following the steps is 48k^8.
  • Combine the coefficient of 48 with the variable k^8.
  • The final expression represents the full solution, with both parts combined seamlessly.
Simplifying expressions consolidates all calculations into a concise, easy-to-read form, providing clarity. It highlights the importance of systematically performing and verifying each step to ensure a correct final result.

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Most popular questions from this chapter

State the conjugate of each binomial. $$-b+2$$

Simplify and write the resulting polynomial in descending order of degree. $$-7 x+5 x^{4}-2+3 x^{4}+7 x+10-3 x^{2}$$

For Exercises, find the GCF of the monomials. \(12 a^{5}\) and \(15 a\)

Complete the table. To find the product in the first row, multiply the two variable expressions. After finding the variable expressions for the product, evaluate each variable expression using the given values. $$\begin{array}{r|c|c|c} & \text { Expression 1 } & \text { Expression 2 } & \text { Product } \\ \hline y & 2 y+3 & 3 y-1 & \\ \hline 10 & & & \\ \hline-1 & & & \\ \hline \end{array}$$

Simplify by combining like terms. $$9 t-12 t$$
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