- The document discusses perfect squares, square roots, cubes, and nth roots.
- It provides examples of finding square roots and cube roots of various numbers.
- Key formulas introduced include the volume of a sphere and estimating distance to the horizon.
4. NOTE : Every positive real number has two real number square roots. The number 0 has just one square root, 0 itself. Negative numbers do not have real number square roots. When evaluating we choose the positive value of a called the principal root . Evaluate Notice, since we are evaluating, we only use the positive answer.
5. For any real numbers a and b , if a 2 = b , then a is a square root of b . Just like adding and subtracting are inverse operations, finding the square root of a number and squaring a number are inverse operations.
6. 2 2 2 x 2 = 4 Perfect Square The square root of 4 is ... 2
7. 3 x 3 = 9 3 3 Perfect Square The square root of 9 is ... 3
8. 4 x 4 = 16 4 4 Perfect Square 4 The square root of 16 is ...
9. 5 5 5 x 5 = 25 Perfect Square Can you guess what the square root of 25 is?
11. This is great, But…. Do you really want to draw blocks for a problem like… probably not! If you are given a problem like this: Find Are you going to have fun getting this answer by drawing 2025 blocks? Probably not!!!!!!
12. It is easier to memorize the perfect squares up to a certain point. The following should be memorized. You will see them time and time again. x x 2 x x 2 0 0 10 100 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 20 400 8 64 25 625 9 81 50 2500
13. To name the negative square root of a , we say To indicate both square roots, use the plus/minus sign which indicates positive or negative.
17. Simplifying Radicals Divide the number under the radical. If all numbers are not prime, continue dividing. Find pairs, for a square root, under the radical and pull them out. Multiply the items you pulled out by anything in front of the radical sign. Multiply anything left under the radical . It is done!
18. Evaluate the following: To solve: Find all factors Pull out pairs (using one number to represent the pair. Multiply if needed)
24. The general rule for reducing the radicand is to remove any perfect powers. We are only considering square roots here, so what we are looking for is any factor that is a perfect square. In the following examples we will assume that x is positive. Gizmo: Simplifying Radicals
27. Examples: E. Unless otherwise stated, when simplifying expressions using variables, we must use absolute value signs. when n is even. *All the sets of “3” have been grouped. They are cubes! NOTE: No absolute value signs are needed when finding cube roots, because a real number has just one cube root. The cube root of a positive number is positive. The cube root of a negative number is negative.
33. N th Roots When there is no index number, n , it is understood to be a 2 or square root. For example: = principal square root of x. Not every radical is a square root. If there is an index number n other than the number 2, then you have a root other than a square root.
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37. Nth Roots Type of Number Number of Real nth Roots when n is even Number of Real nth Roots when n is odd. + 2 1 0 1 1 - None 1
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40. F. Write each factor as a cube. Write as the cube of a product. Simplify. Absolute Value signs are NOT needed here because the index, n, is odd.
43. Evaluate the following: To solve: Find all factors Pull out set’s that contain the same number of terms as the root (using one number to represent the set of 4. Multiply if needed)
46. Solving Equations When solving equations with exponents, you must isolate the variable (with the exponent). Then you must take the appropriate root of both sides of the equation. Since the square and the square root are inverse operations, they cancel each other, as can bee seen on the left side of the equation. To check your solutions: Plug both answers into the original equation. Both answers, 6 and -6, work.
47. Alternative Method If you did, you found that for the equation, , -4 does NOT work!!!!!!! When you plug in -4 for n, you get – 7168, which is not what was given in the equation. So, n = 4 works, n = -4 does not. The solution is n = 4 and the extraneous solution is n = -4. Extraneous solutions do not satisfy original equation and must be discarded. Did you check your answers by plugging both answers into the original equation. Did you check your answers by plugging both answers into the original equation.
49. The following equation is used by ABC Toys to determine how many pieces of a specific round toy will fit into a shipping crate. Find the approximate radius of each toy, rounded to the nearest hundredths, if you know that there are 50 toys in the box. Multiply both sides by 8 to get rid of the fraction. Divide both sides by 3∏ Take the cube root of both sides Round to the correct place value Plug in what you know
