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To fix this we will use the first and second properties of radicals above.

To do this we noted that the index was 2. We then determined the largest multiple of 2 that is less than 7, the exponent on the radicand. To fix this all we need to do is convert the radical to exponent form do some simplification and then convert back to radical form. Although, with that said, this one is really nothing more than an extension of the first example.

There is more than one term here but everything works in exactly the same fashion. We will break the radicand up into perfect squares times terms whose exponents are less than 2 i. Now, go back to the radical and then use the second and first property of radicals as we did in the first example.

This will happen on occasion. So, instead of get perfect squares we want powers of 4. This time we will combine the work in the previous part into one step. That will happen on occasion. Individually both of the radicals are in simplified form.

However, there is often an unspoken rule for simplification. The unspoken rule is that we should have as few radicals in the problem as possible. Performing these operations with radicals is much the same as performing these operations with polynomials. If we are looking at the product of two radicals with the same index then all we need to do is use the second property of radicals to combine them then simplify.

What we need to look at now are problems like the following set of examples. Example 4 Multiply each of the following. It works the same way! Okay, we are now ready to take a look at some simplification examples illustrating the final two rules.

To get rid of them we will use some of the multiplication ideas that we looked at above and the process of getting rid of the radicals in the denominator is called rationalizing the denominator.

In fact, that is really what this next set of examples is about. They are really more examples of rationalizing the denominator rather than simplification examples. Example 5 Rationalize the denominator for each of the following.§ Implementation of Texas Essential Knowledge and Skills for Mathematics, High School, Adopted (a) The provisions of §§ of this subchapter shall be .

Write the piecewise functions for the graph shown. Solution: Step 1: Locate the break point.

Here it is at x = 2. Step 2: Find the equation of the graph to the left of break point. Determine where the graph is positive vs. negative (reflected). Substitute values into the argument to see if the outcome is positive or negative. Derivatives of Absolute Value Functions.

Tutorial on how to find derivatives of functions in calculus (Differentiation) involving absolute value functions.

As an exercise, plot the graph of f and explain the results concerning f'(x) obtained above. Exercises: Find the first derivatives of these functions. A1.N Number and Quantity. A1.N-RN The Real Number System. A1.N-RN.B Use properties of rational and irrational numbers.

A1.N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. The ﬁrst edition of Basic Econometrics was published thirty years ago.

Over the years, there have been important developments in the theory and practice of econometrics. In each of the subsequent editions, I have tried to incorporate the major.

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