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Math  /  Algebra

QuestionRewrite without parentheses. (5x6z5−3x3)(−7xz4)\left(5 x^{6} z^{5}-3 x^{3}\right)\left(-7 x z^{4}\right)
Simplify your answer as much as possible.

Studdy Solution

STEP 1

What is this asking? We need to multiply two expressions and then simplify the result, making it look as clean as possible! Watch out! Don't forget to distribute the second expression to *both* terms in the first expression and be careful with those negative signs!

STEP 2

1. Distribute the terms
2. Simplify the expression

STEP 3

Alright, let's **multiply** the first term, 5x6z55x^6z^5, by −7xz4-7xz^4.
Remember, we multiply the coefficients and add the exponents of the like variables.
So, 5⋅−75 \cdot -7 gives us **-35**.
Then, x6⋅xx^6 \cdot x gives us x6+1=x7x^{6+1} = \textbf{x}^7.
Finally, z5⋅z4z^5 \cdot z^4 gives us z5+4=z9z^{5+4} = \textbf{z}^9.
Putting it all together, we get −35x7z9-35x^7z^9.

STEP 4

Now, let's **multiply** the second term, −3x3-3x^3, by −7xz4-7xz^4.
A negative times a negative gives us a **positive**, so −3⋅−7=21-3 \cdot -7 = \textbf{21}.
Next, x3⋅xx^3 \cdot x gives us x3+1=x4x^{3+1} = \textbf{x}^4.
And we only have z4z^4 in the second term, so it remains as is.
Putting it all together, we get 21x4z421x^4z^4.

STEP 5

Now, we just need to **combine** the results from our previous multiplications.
We have −35x7z9-35x^7z^9 and 21x4z421x^4z^4.
Since these terms don't have the same combination of variables and exponents, we can't combine them any further.
So, our **final simplified expression** is −35x7z9+21x4z4-35x^7z^9 + 21x^4z^4.

STEP 6

−35x7z9+21x4z4-35x^7z^9 + 21x^4z^4

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