Problem 360 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 360

In the following exercises, simplify. $$\left(-2 r^{-3} s^{9}\right)\left(6 r^{4} s^{-5}\right)$$

Short Answer

Expert verified

-12 r s^4

Step by step solution

- Multiplying the coefficients

Multiply the numerical coefficients of the given expressions. In this case, multiply $-2$ and $6$. $-2 \times 6 = -12$.

- Combining like variables

According to the law of exponents $a^m \cdot a^n = a^{m+n}$, add the exponents of the like variables. For $r$-terms:\[ r^{-3} \cdot r^{4} = r^{-3+4} = r^1 = r \]. \For $s$-terms: \[ s^{9} \cdot s^{-5} = s^{9-5} = s^{4} \].

- Express the final simplified form

Combine the results of each variable with the multiplied coefficient: \[ -12 r s^4 \].

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Key Concepts

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

Multiplying Coefficients

When simplifying algebraic expressions, a crucial step is multiplying the coefficients. Coefficients are the numerical parts of a term. In the given problem, the coefficients are $-2$ and $6$. These numbers are multiplied together. The process is simple:

Multiply $-2$ by $6$ to get $-12$.

This gives the new coefficient for our simplified expression. Always remember to multiply the numbers separately from the variables to keep the process clear and organized.

Laws of Exponents

Next, let's talk about the laws of exponents. These laws help combine terms with the same base. In this problem, we deal with two different bases: $r$ and $s$. According to the law of exponents, $a^m \cdot a^n = a^{m+n}$. This means that we can add the exponents when multiplying like bases.

For $r$-terms: $ r^{-3} \cdot r^{4} = r^{-3+4} = r^{1} $.

So, $r^{-3} \cdot r^{4} = r $.

For $s$-terms: $ s^{9} \cdot s^{-5} = s^{9-5} = s^{4} $.

So, $s^{9} \cdot s^{-5} = s^{4} $.
The laws of exponents help simplify expressions involving powers efficiently.

Combining Like Terms

Finally, we need to combine like terms to get the simplified expression. This involves bringing together all parts of the expression after multiplication.

We've multiplied the coefficients to get $-12$.
We've combined the $r$-terms to get $r$ and the $s$-terms to get $s^{4}$.

Bringing all components together, we get:
$-12 \cdot r \cdot s^{4} = -12rs^{4} $.
And there we have it. A fully simplified algebraic expression! Remember, watch out for the coefficients, apply the laws of exponents, and neatly combine the terms at the end.

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91影视

In the following exercises, simplify. $$\left(-2 r^{-3} s^{9}\right)\left(6 r^{4} s^{-5}\right)$$

Short Answer

Step by step solution

- Multiplying the coefficients

- Combining like variables

- Express the final simplified form

Key Concepts

Multiplying Coefficients

Laws of Exponents

Combining Like Terms

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