Problem 133 Multiply the monomials. $$\lef... [FREE SOLUTION]

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Chapter 6: Problem 133

Multiply the monomials. $$\left(6 y^{7}\right)\left(-3 y^{4}\right)$$

Short Answer

Expert verified

-18y^{11}

Step by step solution

- Multiply the Coefficients

Multiply the numerical coefficients of the monomials together: 6 times - 3 equals - 18.

- Apply the Law of Exponents

Add the exponents of the variable y: y^{7+4} y^{11} because according to the rule of exponents, when multiplying like bases, you add the exponents.

- Combine the Results

Combine the results from Step 1 and Step 2: -18y^{11}

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Key Concepts

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

law of exponents

When multiplying monomials, the law of exponents is crucial. This law states that when you multiply powers with the same base, you add the exponents. For example:
If you have $a^m \times a^n$, it will result in $a^{m+n}$.
In our original exercise of $6 y^7 \times (-3 y^4)$, we apply this law to the variable part. Here, both terms have the base $y$ but with different exponents. So, you add the exponents: ${7 + 4 = 11}$, leading to $y^{11}$.
Remember, this rule only works when the bases are the same. For different bases, separate the multiplication and deal with the exponents independently.

coefficients

To solve a problem involving multiplying monomials, you need to handle the coefficients correctly. Coefficients are the numerical parts of the terms. In our example, $6$ and $-3$ are the coefficients. These can be positive or negative numbers.
The rules of multiplication for the coefficients are straightforward:

Multiply the numbers as you usually would.
If both numbers are negative or positive, the result is positive.
If one number is negative and the other is positive, the result is negative.

So, in the given exercise:
$6 \times (-3) = -18$. It鈥檚 straightforward arithmetic!

combining like terms

Combining like terms means simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. However, when multiplying monomials, you don't necessarily combine like terms because you are not adding or subtracting.
Nevertheless, it's key to identify and simplify terms with the same base and exponent:

In multiplicative forms, such as $y^a \times y^b$, you apply the law of exponents to combine them into one term with a new exponent.
Always ensure the final result maintains its simplest form.

In our problem, after multiplying the coefficients and applying the law of exponents to the variables, you combine the results to get the final expression:o $-18y^{11}$
This is the simplified form of the multiplied monomials.

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91影视

Multiply the monomials. $$\left(6 y^{7}\right)\left(-3 y^{4}\right)$$

Short Answer

Step by step solution

- Multiply the Coefficients

- Apply the Law of Exponents

- Combine the Results

Key Concepts

law of exponents

coefficients

combining like terms

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