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Radical Expressions and Functions

Radical Expressions and Functions. Section 8.1 MATH 116-460 Mr. Keltner. Finding the n th root of a number. Finding the square root of a number involves finding a number that, when squared, equals the given number. In other words, finding such that b 2 = a .

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Radical Expressions and Functions

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  1. Radical Expressions and Functions Section 8.1 MATH 116-460 Mr. Keltner

  2. Finding the nth root of a number • Finding the square root of a number involves finding a number that, when squared, equals the given number. • In other words, finding such that b2 = a. • Some vocabulary involved with nth roots: This is called a radical symbol. n is the indexof the expression. The index tells us what amount of factors we should look for in order to simplify a quantity. Examples: If n = 3, we are looking for some value r such that r3 = s. If n = 4, we are looking for some value r such that r4 = s. s is called the radicand of the radical expression. If the index n is even, then s must be positive. This is because there is no value of r such that r2 = -s.

  3. Nth roots: generally speaking • We can define the nth root of a number as this: • b is the nth root of a number a if bn = a. • The principal root of a nonnegative number is just the nonnegative root. • Example: The square root of 16 could be 4 or -4, but it is easiest to find a single root, so we usually only indicate the principal root. • The only exception would be solving an equation, where either the principal root or the secondary root could both supply solutions to the equation (that is why it becomes important to check your answers).

  4. Evaluating nth roots • When evaluating a radical expression like the sign of a and the index n will determine possible outcomes. • If a is nonnegative: • Then , where b ≥ 0 and bn = a. • If a is negative: • And n is even, then there is no real-number root (no real solutions). We will cover these expressions in section 8.7. • And n is odd, then , where b is negative and bn = a.

  5. Example 1: Evaluating Nth roots • Simplify each of the following roots. • Click each expression to reveal the answer. 0.6 -2 ±13 No real root 5 6 4 -3 -3

  6. Irrational Roots • All the examples we have seen so far have been rational solutions (those that are whole numbers or decimals that terminate or have some repeating pattern, like 1/3 or 2/11. • Some roots, like , are called irrational, because they do not have a decimal that ends or has a repeating pattern. • Another example of an irrational number is π, which we usually just round to 3.14. • We cannot express the exact value of this expression without rounding error of some sort. • So, is actually considered the exact value of .

  7. Evaluating Roots with a Calculator • When we graphed polynomial functions, it became evident that our graph was more accurate by plotting additional points on the graph. • When evaluating roots on a calculator, our answer becomes more accurate with more decimal places. • This table illustrates how accurate our estimate of becomes by using more and more decimal places.

  8. Nth Roots on a Calculator • Evaluating nthroots on a calculator can be easy, but you must know where to look. • Using the MATH key (just below the ALPHA key), you can evaluate any nth root you wish and the calculator will return the principal root. Follow these steps: • Enter the index of the expression, • Press the MATH key, • Then 5:x√( and enter the value you want to evaluate. • These steps are very similar for several models of calculators. Try these:

  9. Simplifying Radical Expressions Definition of roots Power of a Power The Power of a Power property of exponents tells us that (am)n = amn. We will use this property to verify roots in the next examples, as well as to help us simplify irrational roots. By writing an original quantity as a perfect square, it may help simplify the square root of the quantity. • Remember that, when the index is even, the principal root will be nonnegative. • This allows us to assume that all variables are represented by nonnegative values.

  10. 10x2 9y6 Example 3: Simplifying Expressions • Simplify each of the following roots. • Click each expression to reveal the answer. x2 3x5 b2 6a6 3y3

  11. Radical Functions • Radical functions are those that are defined by a radical expression. • From section 3.1, we will see them in function notation, usually as f(x) or similar notation. • This is, essentially, a fancier way of writing “y.” • To evaluate for a particular value of x, such as 2, the notation will simply look like f(2). • We do not have to divide the equation when we are done, just substitute the given value for the variable and simplify.

  12. Domain of Radical Functions • With even indexes, it is not possible to evaluate the root of a negative value. • Because of this, we must only use input values that result in nonnegative outputs. • To evaluate the domain in such cases, set the radicand greater than or equal to zero (≥0) and solve to get the domain of the function.

  13. Example 4: Domain of Radical Functions • Evaluate the domain of each radical function. • Click the function to see the answer. -2 ≤ x ≤ 2 x ≥ 4

  14. Assessment Pgs. 541-543: #’s 9-108, multiples of 3 (most are calculator problems)

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