Problem 98 \(\left(2 y^{-2}\right)\left(6 y... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 5: Problem 98

$\left(2 y^{-2}\right)\left(6 y^{3}\right)$

Short Answer

Expert verified

\(\left(2 y^{-2}\right)\left(6 y^{3}\right) = 12y^1 = 12y\

Step by step solution

Analyze the Terms

We look at the expression $\left(2 y^{-2}\right)\left(6 y^{3}\right)$. We consider these as two terms, 2y^-2 and 6y^3. We can separate the constants and the powers of y as follows: 2*6 and y^-2*y^3.

Multiply the Constants

We multiply the constants 2 and 6 together to get 12.

Apply the Law of Exponents

With the y terms, y^-2 and y^3, use the rules of exponents for multiplication and add the exponents. The exponent of the resulting y term would be -2+3, which is 1

Combine the Constants and Powers of y

Finally, we combine the results from steps 2 and 3 to get the final answer. So, we get 12*y^1

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Key Concepts

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

Multiplying Exponents

When you multiply terms with exponents, the process is simpler than it might seem! Imagine you have two terms,

$2 y^{-2}$ and $6 y^{3}$.

To handle them, start by separating the numbers (constants) from the variables with exponents.
The constants are 2 and 6, while the variable factor involves $y$ raised to different powers. Multiply the constants first:

$2 \times 6 = 12$

Then tackle the exponents, which we鈥檒l discuss using the law of exponents next.

Negative Exponents

Negative exponents might look strange at first, but they follow a simple rule. An expression like $y^{-2}$ is the same as $\frac{1}{y^{2}}$. It's like flipping the base over.
So if we see $y^{-2}$, it means we're dealing with the reciprocal of $y^{2}$.
Handling them during multiplication becomes straightforward when you remember this key concept! When multiplying $y^{-2}$ by $y^{3}$, you simply add the exponents, keeping this flip in mind.

Law of Exponents

The law of exponents is essential for working with exponents. It says that when you multiply two powers that have the same base, you can add the exponents together.
For example, when dealing with $y^{-2}$ and $y^{3}$, you simply add the exponents:

-2 + 3 = 1

This turns the expression into $y^{1}$, which is just $y$.
The rule makes operations smooth and efficient, so always keep it handy!

Polynomial Multiplication

Polynomial multiplication involves applying the same rules to each variable and constant. For our example,

$\left(2 y^{-2}\right)\left(6 y^{3}\right)$

we separate constants and variables, multiplying each part individually.
First, the constants:

Multiply them: $2 \times 6 = 12$

Next, the variables with exponents:

Add the exponents: $-2 + 3 = 1$

Finally, combine them to get the result $12y^{1}$, or simply $12y$. By breaking down each step, polynomial multiplication becomes a lot easier to manage!

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91影视

\(\left(2 y^{-2}\right)\left(6 y^{3}\right)\)

Short Answer

Step by step solution

Analyze the Terms

Multiply the Constants

Apply the Law of Exponents

Combine the Constants and Powers of y

Key Concepts

Multiplying Exponents

Negative Exponents

Law of Exponents

Polynomial Multiplication

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