The product rule for exponents states that for two or exponents that share the same base, the products is equal to their shared base to the sum of their exponents. Can anyone help me figure this out?

10⁻⁵ · 10⁴ · 10⁸ = 10⁻⁵⁺⁴⁺⁸ = 10⁷

105 = 10 x 10 x 10 x 10 x 10 = 100,000 (5 zeros)

108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000 (8 zeros)

Notice that the number of zeros in the ordinary decimal expression is exactly equal to the power to which 10 is raised.

If the number is expressed in words, first write it down as an ordinary decimal number and then convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, so in powers of ten notation ten million is written 107.

A number which is some power of 1/10 can also be expressed easily in scientific notation. By definition,

1/10 = 10-1 ("ten to the minus one power")

More generally, the expression "10-n" (where n is a whole number) means ( 1/10 )n. Thus

10-3 = ( 1 / 10 )3 = 1 / ( 10 x 10 x 10) = 1/1000

10-8 = ( 1 / 10 )8 = 1/100,000,000

The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut! The "power rule" tells us that to raise a power to a power, just multiply the exponents

Step-by-step explanation:

the revenue of the fitness center, y, is the product of two factors: the number of members and the fee paid by each member. there are 320 members now, but number of members goes down by 10 for each $5 increase in fees, so if x is the number of $5 price increases then members is equal to (320 - 10x). since the fee is currently $45, and the fee increases by $5 for every price bump x, the fee is given by (45 + 5x). so, revenue y is given by y = (320 - 10x)(45 + 5x).