SOLUTION: Explain why -5x² and (-5x)² are not equivalent. Explain why the expressions 2³ × 5³ and 10³ are equivalent.

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Question 630183: Explain why -5x² and (-5x)² are not equivalent.
Explain why the expressions 2³ × 5³ and 10³ are equivalent.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Important: Exponents only apply to what is literally in front of them!

In -5x%5E2 it is the "x" that is in front of the exponent. So the exponent applies only to the x. So -5x%5E2 is short for -5%2Ax%2Ax.

In %28-5x%29%5E2 it is the right parenthesis that is in front of the exponent. This means that the exponent applies to the whole expression inside the parentheses. So %28-5x%29%5E2 is short for %28-5x%29%2A%28-5x%29 (which simplifies to 25x%5E2).

One of the rules/properties of exponents is:
a%5En%2Ab%5En+=+%28a%2Ab%29%5En
The expression
2%5E7%2A5%5E7
matches the pattern of the left side of the rule above. So according to the rule it must be equal to the right side of the rule:
%282%2A5%29%5E7
which simplifies to
10%5E7

As usual, if you have trouble using the rules for exponents, work without them! In other words, rewrite your expressions without exponents and simplify without them.
2%5E7%2A5%5E7
becomes
2*2*2*2*2*2*2*5*5*5*5*5*5*5
This is all multiplication so we can use the Commutative Property of Multiplication to change the order as we please and we can use the Associative Property of Multiplication to rearrange the grouping as we please. So we can change the above to:
(2*5)*(2*5)*(2*5)*(2*5)*(2*5)*(2*5)*(2*5)
which simplifies to:
(10)*(10)*(10)*(10)*(10)*(10)*(10)
which can be rewritten as
10%5E7