0.4 Nth Roots and Real Exponents

Bell Ringer Which of the following lists all the roots of

0.4 Nth Roots and Real Exponents Yesterday finished up our study on Quadratics. Today we will begin our study on exponential functions by investigating the properties of real exponents. We will: Find nth roots Simplify using absolute value Use the properties of exponents Evaluate and simplify expressions containing rational exponents. Solve equations containing rational exponents Tomorrow we will be solving systems of linear equations .

Real nth roots of Real Numbers

Simplify = = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM = = = = = =

GROWING A FACTOR TREE

180 18 10 Can we grow a tree of the factors of 180? Or You might notice that 180 has a ZERO in its ONES PLACE which means it is a multiple of 10. SO… 10 x  = 180 10 x 18 = 180 Or You might notice that 180 has a ZERO in its ONES PLACE which means it is a multiple of 10. SO… 10 x  = 180 180 You might see that 180 is an EVEN NUMBER and that means that 2 is a factor… 2 x  = 180 ? Can you think of one FACTOR PAIR for 180 ? This should be two numbers that multiply together to give the Product 180. Can we grow a tree of the factors of 180? 10 18

You have to find FACTOR PAIRS for 10 and 18 180 NOW You have to find FACTOR PAIRS for 10 and 18 We “grow” this “tree” downwards since that is how we write in English (and we can’t be sure how big it will be - we could run out of paper if we grew upwards). 10 18

180 Find factors for 10 & 18 18 10 2 x 5 = 10 6 x 3 = 18 5 2 6 3

5 10 2 6 180 18 3 Since 2 and 3 and 5 are PRIME NUMBERS they do not grow “new branches”. They just grow down alone. Since 6 is NOT a prime number - it is a COMPOSITE NUMBER - it still has factors. Since it is an EVEN NUMBER we see that: 6 = 2 x  ARE WE DONE ??? 2*3=6 2 5 3 2 3

… and if we flip it over we can see why it is called a tree 2 3 5 10 6 180 18 … and if we flip it over we can see why it is called a tree

Examples Simplify each expression: a) b) c)

Simplify the expression: Your Turn Simplify the expression:

Simplify Using Absolute Value Simplify Use division. Divide the index number into the exponent. The remainder goes under the radical. Any time you take out an odd exponent with an even index number, absolute value symbols are required.

Simplify. Examples

Simplifying Using Absolute Value Your Turn

Motivating the Lesson Property used Use the properties of exponents to find a number x such that Property used

Simplify each expression. Examples

Simplify each expression. Your Turn Simplify each expression.

State whether represent the same quantity. Explain. Think about it…….

Simplify each expression. Examples b. c. a.

Simplify the expression. Your Turn

Simplify each expression. Examples c. a. b.

Simplify the expression. Your Turn Simplify the expression.

Simplify the expression. Examples

Simplify the expression. Your Turn Simplify the expression.

Examples a. Express using rational exponents. b. Express using a radical.

If au = av, then u = v Example This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal. Example The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something? Now we use the property above. The bases are both 2 so the exponents must be equal. We did not cancel the 2’s, We just used the property and equated the exponents. You could solve this for x now.

The left hand side is 4 to the something but the right hand side can’t be written as 4 to the something (using integer exponents) Let’s try one more: We could however re-write both the left and right hand sides as 2 to the something. So now that each side is written with the same base we know the exponents must be equal. Check:

Your Turn Solve

HOT Question If 5-2 is raised to the third power, Determine whether the statement makes sense or does not make sense, and explain your reasoning. If 5-2 is raised to the third power, the result is between 0 and 1.

Wrap-up How do you simplify a radical? Which is correct:

Practice makes perfect! 0.4 Practice Practice makes perfect!