Math 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick

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1 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick Math Chapter 1 Review Exponents and Radicals Exponent definitions and rules: For the expression, is the exponent and is the base. means =8 1 Definition: exponential form for expressing a root x a = ( x a ) b = x ab x a x a x b = x a+b x b = xa b (xy) a = x a y a a x 0 " x =1 = x a y & y a x a = 1 x a a " x Note also that = y a " y & x & Warning: (x + y) a cannot be simplified by any exponent rule but can be multiplied out if necessary or left as is. a x No calculator is needed for any of these problems. Carry out the indicated operations and simplify as much as possible. If a numerical base has an exponent is or less, express the number without an exponent. A x x 4. b b 4 5. x y 11 x y 1 6. ( ) 4 7. ( ) 8. () ( ) 7 1

2 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick ( 8) " 7 64 & 15. " 7 64 & " & 0 B. 1. ( ). (x 6 ) 4. (m ) 5 4. (z ) 6 5. (t 11 ) 6. (6a 4 ) (a) 7. d (d 7 ) 5 8. (abc ) (a b) C. 1. $ & ". " 4 &. " x 4 y 5 $ & 4. " x $ & 5. m $ & " n x 5 y 4 $ " 5z & 7. " t 40 h 10 $ & 8. y $ " z &

3 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick (8 ) 10. (7 ) 11. " x y & 1. 5a & ( $ 10a ' 1 D. 1. x 5 x 8 x7. x 11. y 1 y 4. 5x4 h 7 5x 5 h 5. (r ) 4 (r ) 8 6. m m 5 m 7. x 5 1 x 8 x 8. y x6 y 11 E. Evaluate (write without exponents) F. Rewrite so that there are no negative exponents by moving expressions up and down. Then simplify. 1. x 4 y 5 zq (Note that the does not move). x x 6 x x 6 x 7

4 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick Radical definitions and rules: These are really the same rules as the exponent rules. It is often easier to work with exponents rather then radicals using the definition x 1 a a = a ab = a b b = a b Warning: a + b cannot be simplified unless the numbers a and b be combined in some way. Conventions for simplifying radicals: When possible combine roots to have a single root. Extract perfect squares from roots. Do not leave roots in denominators. x G x x 7. 7x 8. 8x y 6 z x y

5 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick H. Problems with geometric figures. 1. Find the area and perimeter of the rectangle shown. 5 cm 5 cm. Find the area and perimeter of the right triangle. ft 4 ft. The area of a circle is. Find the radius and circumference of the circle. 4. For each right triangle, find the length of the third side. Write your result in simplified form. a. Legs of 8 in and 10 in. 5

6 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick b. legs of inch and inch c. a leg of inch and a hypotenuse of inch. I. Use the rules of exponents to solve each equation. (Hint: rewrite both sides of the equation as exponents with the same base.) 1. = 7. = 81. = 9 4. ( )( ) = 7 5. = 7 6. = 81 J. Find a function that fits each table. 1. x f(x) x f(x)

7 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick Rational and Irrational Numbers Rational: a number is rational (Q) if it can be written as a fraction composed of integers. Symbolically, x Q (x is an element of the set of rational numbers) if x = where a Z (a is an integer) and b Z, b 0 (b is an integer not equal to zero.) Irrational: irrational numbers that we are familiar with include square roots of non-perfect squares and π (pi). Theorems About Rational and Irrational Number A rational number plus an irrational number is irrational. (Ex: + 5 is irrational) A non-zero rational number times an irrational number is irrational. (Ex: 5 is irrational) Square roots of prime numbers are irrational. You can show that a number is irrational by using one of the theorems above. You can show that a number is rational by writing it as a fraction composed of integers (make sure your fraction is equal to the original number). K. Determine whether each of the following numbers are rational or irrational. Give a reason for your answer

8 Math : Algebra, Geometry and Statistics Ms. Sheppard-Brick L. Put the following number sets in order from least to greatest. Use inequality symbols or equal symbols as appropriate to separate the numbers. (Hint: simplify first.) 1. 4, 4, 16,, ,, 50, 5, ,,,, 8