## Exponentials Solutions

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Exponentials Solutions: Problem 1 :: Problem 2 :: Problem 3

**PROBLEM 1:** (top)

__Problem:__ Simplify the exponential expression .

__Solution:__ First, use the invert and multiply rule to change the division into multiplication, getting . Now, use the division rule to remove the second quotient in the expression, forming . Next, get rid of the exponents outside parentheses with the power rule: . Use the multiplication rule to combine the two terms to get . Finally, use the divison rule to cancel out like terms, then simplify to get the final answer of .

**PROBLEM 2:** (top)

__Problem:__ Simplify the exponential expression .

__Solution:__ Start by converting 625, 125, and 25 to the appropriate powers of 5: . Next, use the power rule to get rid of exponents outside of parentheses, yielding . Now use the multiplication rule to combine terms in the denominator . Notice that there is a power of 4x in every term, both in the numerator and the denominator. We'd like to cancel this out, so we'll use the multiplication rule in reverse to separate these powers. This gives us . We can use the distributive property to rewrite the denominator like this, , then cancel the 5^{4x} terms to find . At this point, it may appear that we're done, and this answer is acceptable; since there is no rule to deal with addition of like bases, we can't directly manipulate the denominator. It would be better to get rid of the negative exponent in the denominator, however. We can do this by using the distributive property again, pulling out the 5^{-6} term in front, getting

**PROBLEM 3:** (top)

__Problem:__ Graph the function .

__Solution:__ The base is positive, so this is an exponential growth function. It's initial value is negative, though, so the graph lies entirely below the x-axis and the y-intercept is at (0, -2). The graph is shown below.

This page last updated 16 June, 2008 11:32 AM