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Chapter 5: Problem 3

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

Short Answer

Expert verified
15 y^{11}

Step by step solution

01

Identify the coefficients

Identify and multiply the numerical coefficients from each term. Here, the coefficients are 5 and 3. Multiply them together: 5 脳 3 = 15
02

Identify the variables and their exponents

Identify the variable 'y' in each term and note their exponents. In this case, the exponents are 4 and 7.
03

Add the exponents

Add the exponents of the variable 'y' since they are being multiplied: 4 + 7 = 11
04

Combine the results

Combine the coefficients and the variable with the new exponent: 15 y^{11}

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Key Concepts

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

Understanding Coefficients
When multiplying polynomials, we first identify the coefficients of each term. Coefficients are the numerical parts of the terms. For example, in the term \(5 y^{4}\), the coefficient is 5. Similarly, in the term \(3 y^{7}\), the coefficient is 3.
Now, multiply these coefficients together: 5 脳 3 = 15.
This result becomes the new coefficient of the product.
Working with Exponents
In this multiplication problem, we also need to handle the exponents of the variable y. An exponent tells us how many times the base (in this case, the variable y) is used as a factor.
For instance, in the term \(5 y^{4}\), the exponent is 4, meaning y is multiplied by itself 4 times (y 脳 y 脳 y 脳 y).
To find the product of \(y^{4} \times y^{7}\), add the exponents together: \4 + 7 = 11\.
This sum becomes the new exponent of y, giving us \(y^{11}\).
Combining Like Terms
Finally, to combine the results, we merge both the new coefficient and the new exponent.
We already found that the new coefficient is 15 and the new exponent of y is 11.
Therefore, the product of \(5 y^{4} \times 3 y^{7}\) is \15 y^{11}\.
Combining like terms involves adding or subtracting the coefficients of terms with the same variables and exponents. However, in this case, since we are multiplying, we simply combine the coefficient and the product of the variables and exponents together.

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Most popular questions from this chapter

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