Any questions on the Section 6.6 homework?

Any questions on the Section 6.6 homework?

Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note-taking materials.

Section 7.1 Radicals and Radical Expressions

Today’s Homework Assignment: Don’t panic when you see that there are 49 problems on today’s homework assignment on section 7.1. Most of them are quick, simple problems like finding the square root of 25 or the cube root of 8. In previous semesters, this has been one of the shortest homework assignments of the semester in terms of the time spent to complete it.

Square Root • Opposite of squaring a number is taking the square root of a number. • A number b is a square root of a number a if b2 = a. • In order to find a square root of a, you need a number that, when squared, equals a. • NOTE: For many square and cube root problems, you may find it faster to use the list of perfect squares and cubes on your formula sheet than to use your calculator. (If you need an extra copy for your notebook, get it now.)

The principal (nonnegative) square root is noted as The negative square root is noted as

A radical expression is an expression containing a radical sign. • A radicand is the expression under a radical sign. • Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

Examples: How would you check this answer? not a real number (there’s no real number that gives -4 when squared.)

Problem from today’s homework:

Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). • Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers. • IF REQUESTED in the problem statement you can find a decimal approximation for these irrational numbers using your calculator. • Unless an approximation is requested, leave answers in radical form. (This is also referred to as the “exact answer”.) • Do not convert to an approximation (decimal form) unless explicitly requested to do so.

Example Radicands might also contain variables and powers of variables. Simplify. Assume that all variables represent positive numbers. How would you check this answer? Check?

The cube root of a real number a Note: a is not restricted to non-negative numbers for cubes. The cube root of a negative number is a negative number. Think about this: What is (-2)3? Answer: -8 Therefore

Example (You can use your formula sheet to find the cubed numbers) How would you check this answer? Check? Check?

Other roots besides square and cube roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number when a is negative.

Example Simplify the following. Assume that all variables represent positive numbers. Check? Check?

Problem from today’s homework: Check?

Problem from today’s homework: Check? What if the problem looked like this?

If the index of the root is even, then the notation represents a positive number. But we may not know whether the variable a is a positive or negative value. Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

If n is an even positive integer, then If n is an odd positive integer, then

Example Simplify the following. If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign. i.e. if given that a > 0, then we can state that

Problem from today’s homework: How would your answer differ if the instructions said “Assume that x is a real number?”

Example Simplify the following. Since the index is odd and we do not know whether the variables a or b are positive or negative, a and b3 could represent either positive or negative numbers. Therefore, could represent either a positive or a negative number.

Problem from today’s homework: What if this question asked you to find f(-1)? How about f (-2)?

Since every value of x that is substituted into the equation produces a unique value of y, the root relation actually represents a function. The domain of the root function when the index is even, is all nonnegative numbers. (How would you write this in interval notation?) The domain of the root function when the index is odd, is the set of all real numbers. (How would you write this in interval notation?)

We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape. You should have a basic familiarity with root functions, as well.

Example Graph y xy 6 4 2 (6, ) (4, 2) (2, ) 2 x (1, 1) 1 1 (0, 0) 0 0

Now, what is the domain of this function? (in other words, for what values of x is it defined?) The square root function is only defined for values of x that make the radicand (the number under the radical) ≥ 0. So for the function , the domain is { x | x ≥ 0} (in set notation) What would this be in interval notation? Answer: [0, ∞)

Go back to this problem, graph f(x) and find the domain:

Example Graph y xy 8 2 4 (8, 2) (4, ) x (1, 1) -1 1 1 -1 (-1, -1) (0, 0) 0 0 (-4, ) (-8, -2) -4 -8 -2

Now, what is the domain of this function? (in other words, for what values of x is it defined?) The cube root function is defined for ALL values of x. So for the function y = , the domain is { x | x is a real number} What would this be in interval notation? Answer: (-∞, ∞)

You may now OPEN your LAPTOPS and begin working on the homework assignment.

Any questions on the Section 6.6 homework?