/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 14

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

Short Answer

Expert verified
-21t^{17}

Step by step solution

01

- Multiply the Coefficients

First, multiply the numerical coefficients: \(7 \times 3 \times (-1)\). This gives: \(7 \times 3 = 21\) and then multiply by \(-1\): \(21 \times (-1) = -21\).
02

- Add the Exponents of like Terms

Next, add the exponents of the variable \(t\): \(t^5\), \(t^4\), and \(t^8\) have exponents 5, 4, and 8 respectively. Add these exponents: \(5 + 4 + 8 = 17\).
03

- Combine the Results

Finally, combine the results from the previous steps: The coefficient is \(-21\) and the exponent is \(t^{17}\). So the final product is: \(-21t^{17}\).

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Key Concepts

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

Multiplying Coefficients
Multiplying coefficients is the first step in polynomial multiplication. Let's use the exercise example with the coefficients 7, 3, and -1. Coefficients are the numerical parts of a term and can be multiplied together just like regular numbers. Start by multiplying 7 and 3, which gives 21. Then, multiply this result by -1 to get -21. So, when multiplying coefficients, simply perform the multiplication step-by-step to get your final numerical result.
Adding Exponents
Adding exponents is essential when multiplying terms with the same base. In our exercise, the base is the variable 't'. The rule for multiplying exponents states: when you multiply powers with the same base, you add the exponents. Consider our variable 't' with exponents 5, 4, and 8. Adding these exponents together: 5 + 4 + 8, we get 17. So, the combined exponent for variable 't' after multiplication is 17.
Polynomial Multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial and then combining like terms. Our exercise is a simplified version, focusing on terms rather than polynomials. We multiply the coefficients and add the exponents, then we combine the results. Starting with the coefficients 7, 3, and -1, we get -21. For the variable 't', adding the exponents 5, 4, and 8 gives us 17. Therefore, the product of multiplying these terms is: \(-21t^{17}\).

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