Question 471891: Simplify using PRODUCT rule :
(-2x)3(-2x3)
I got 2 answers. -8x5 and 4x5
please explain steps!
Answer by karaoz(32)   (Show Source):
You can put this solution on YOUR website! (-2x)3(-2x3)
First, we can see the above expression as the product between two factors: (-2x)3 and (-2x3).
Both factors are rather complex and we should first make sure we understand what each one of them stands for.
The first factor is (-2x)3 and since the power 3 is on top of the parentheses then the whole thing inside the parentheses need to be raised to the power of 3.
What we have inside the parentheses is the product of two factors, -2 and x. Whenever the product of two or more factors is raised to a certain power, the whole thing can be represented as the product of each of the factors raised to that power.
In other words, (ab)n = anbn. This is also true for any number of factors, for example, (abcxz)n = anbncnxnzn. So, our first factor then can be rewritten as
(-2)3x3, which is equivalent to -8x3 since (-2)3 = (-2)(-2)(-2) = -8.
Going back to the original expression, after we simplified the first factor, we now have:
(-2x)3(-2x3) = -8x3(-2)x3,
which is the product of 4 factors: -8, -2, x3 and x3.
The product between -8 and -2 is 16, while the product between x3 and x3 is x6. This last result is obtained by using PRODUCT rule for exponents, which states that for any m and n (xmxn=xm+n).
Hence,
(-2x)3(-2x3) = 16x6
By the way, PRODUCT rule for exponents can be understood by realising that xm is in fact x multiplied by itself m times, while xn is x multiplied by itself n times. Therefore, the product xmxn is in fact x multiplied by itself m times, which is further multiplied by x multiplied by itself n times. Taken all these products at once, we can see that what we have is x multiplied by itself m+n times, which can be written succinctly as xm+n. This is why xmxn = xm+n
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