uc3rmw_0301e01

Hi there. My name is Paul, and we're going to evaluate exponential functions at different input values. This example has four parts. Let's take a look at the first part. Part A says, let f of x be equal to 3 to the power x minus 2. We want to evaluate f of x at x equal 4. We substitute 4 in for x, so we have that f of 4 is equal to 3 to the power 4 minus 2, having replaced x with 4 in the exponent. We simplify the exponent. We have 4 minus 2 is 2, and so we're evaluating 3 squared, which is 9. And so our solution is that f of 4 is equal to 9. Now we move on to part B. We want to evaluate the function g of x equal minus 2 times 10 to the x at x equals minus 2. So as we did in part A, we'll replace 6 with minus 2. We have g of minus 2 is equal to minus 2 times 10 to the minus 2 power. The exponent is negative, so we're going to take the reciprocal of that factor to convert it to a factor with a positive exponent. So we have minus 2 times 1 over 10 to the second power. Note that the minus 2 out front is not part of the exponent factor in the expression. As a positive exponent, the factor is now easy to evaluate. 10 squared is 100. So we have minus 2 times 1 over 100, and that's equal to minus 2 over 100 or minus 0.02. Therefore, g of minus 2 is equal to minus 0.02. Let's take a look at part C. Here we want to evaluate the function h of x equal 1/9 to the x power at x equal minus 3/2. We replace x with minus 3/2, and we have that h of minus 3/2 is equal to the fraction 1/9 to the power minus 3/2. So we note that the base of this exponent is the reciprocal of 9, so we're going to express that as an exponent 9 to the negative 1 power-- and all that taken to the minus 3/2 power. At this point, we're going to invoke the property of exponents, where when we have something taken to a power and then that whole thing taken to another power, we go ahead and multiply the exponents. So we have 9 to the minus 1 times, in parentheses, minus 3/2. Multiplying those two exponents together, we have 9 to the 3/2 power. At this point, we can go ahead and split off the 1/2 power, take that first. The 1/2 power becomes radical 9, and all of that is being taken to the third power. Radical 9, or square root of 9, is simplified to 3, and that's taken to the third power. 3 to the third power is 27. So we have that h of minus 3/2 is equal to 27. So in the last part of this example, in part D, we want to evaluate f of x equal to 4 to the x power at x equals 3.2. So f of 3.2. We'll go ahead and replace x with 3.2, so we have 4 to the 3.2 power. In this case, there are no exponent properties that we can apply to manipulate this expression to a form that will allow us to evaluate this expression without a calculator. So we go ahead and plug this into our calculator, and we get that 4 to the 3.2 power is approximately 84.44850629, out to eight digits past the decimal point. So that's it. We just evaluated exponential functions by using the laws of exponents and also with a calculator.