5^3 over 5^7 please help, step by step?

1 / 5^4  (or 1 / 625)

Step-by-step explanation:

The easiest way to understand and solve this is by expanding everything:

5^3 = 5*5*5

5^7 = 5*5*5*5*5*5*5

Now put those over each other like this:

        5*5*5

5*5*5*5*5*5*5

Then I essentially draw "1"s or lines over each pair of 5s (from the numerator to the denominator) until all of the 5s on the numerator or denominator are used up.

Note: This is completely legal as you are simply dividing pairs of 5/5 to get many 1s, and if you multiply everything by 1, it will eliminate the 1s.

So once you "cross all of them out" you are left with a 1 at the top and 5*5*5*5 at the bottom.

So you can simplify that down to

1 / 5^4  (or 1 / 625, however it's usually not necessary to calculate larger exponents)

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

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x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

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x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

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x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!