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Chapter 4: Problem 18

Find each product. $$ \left(7 t^{5}\right)\left(3 t^{4}\right)\left(-t^{8}\right) $$

Short Answer

Expert verified
The product is \[-21t^{17} \] .

Step by step solution

01

- Multiply the coefficients

Begin by identifying and multiplying the coefficients of the terms: 7, 3, and -1. Multiply these together: \[ 7 \times 3 \times (-1) = -21 \]
02

- Add the exponents of the same base

Since all the terms have the same base \( t \), add the exponents together: \[ 5 + 4 + 8 = 17 \]
03

- Write the final product

Combine the coefficient from Step 1 with the base and the exponent from Step 2: \[ -21t^{17} \]

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Key Concepts

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

Product of Coefficients
When you multiply polynomials, you should start with the coefficients. Coefficients are the numerical part of a term. For example, in the term \(7t^5\), 7 is the coefficient. In the given exercise, we have coefficients 7, 3, and -1. By multiplying these, we get:
\[7 \times 3 \times (-1) = -21\]
Remember that multiplying positive and negative numbers works as follows:
  • Positive \( \times \) Positive = Positive
  • Negative \( \times \) Negative = Positive
  • Positive \( \times \) Negative = Negative
So, our result \(-21\) is negative because we multiplied three numbers together, where one was negative.
Polynomial Exponents
Next, let鈥檚 understand how to handle the exponents in polynomials. In our example, the exponents are 5, 4, and 8. These exponents are based on the variable \(t\).
When multiplying terms with the same base, simply add the exponents together. This is because of the exponent rule: \[x^a \times x^b = x^{a + b}\].
Applying this rule to our terms, we have:
\[t^5 \times t^4 \times t^8 = t^{5 + 4 + 8} = t^{17}\].
Adding exponents ensures that the variable's power increases correctly. Always make sure all terms share the same base variable when you do this.
Multiplying Monomials
Combining what we have learned so far, let鈥檚 multiply these monomials completely. A monomial is simply a term made up of a coefficient, a variable, and an exponent. Here鈥檚 our given problem:
\((7 t^5)(3 t^4)(-t^8)\).
We already multiplied the coefficients (\text{-21}) and added the exponents (\text{17}):
  • Step 1: Multiply 7, 3, and -1 to get -21
  • Step 2: Add exponents 5, 4, and 8 to get 17
  • Step 3: Combine them into one term
This gives us the final product:
\[-21t^{17}\].
Understanding these steps ensures that you can multiply any set of monomials confidently and accurately.

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Most popular questions from this chapter

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}+2\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

$$\text { Subtract the sum of } 9 t^{3}-3 t+8 \text { and } t^{2}-8 t+4 \text { from the sum of } 12 t+8 \text { and } t^{2}-10 t+3$$

Find each product. $$ -5 t^{4}\left(2 t^{4}+1\right)\left(2 t^{4}-1\right) $$

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 75\. \(\frac{0.00000072(0.00023)}{0.000000018}\)

Find each product. $$ (3 x+2 y)(5 x-3 y) $$
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