Drill #37 Factor the following polynomials Simplify. Find the value of r:

Drill #38 Factor the following polynomials Simplify.

Drill #60 Factor the following polynomial Evaluate the following roots:

7-4 Nth Roots Objective: To simplify radicals having various indices, and to use a calculator to estimate the roots of numbers.

Square Roots What power is a square root? A square is the inverse of a square root…

Square Root* Definition: For any real numbers a and b, if then a is a square root of b or We can also write square roots using the ½ power.

Cube Root* Definition: For any real numbers a and b, if then a is a cube root of b or We can also write cube roots using the 1/3 power.

nth Root* Definition: For any real numbers a and b, if then a is a nth root of b or We can also write nth roots using the power.

How many ways can you … Multiply two numbers to get a positive #? Multiply two numbers to get a negative #?

Examples: Roots (of powers of 2) Even Roots: Odd Roots:

Principal Root* Definition: When there is more than one real root of a number (even numbered roots), the non-negative root is the principal root  Principal Root  Negative Root  Both

Examples: Roots (of powers of 2) Both Principal Negative Roots: Roots: Roots:

Roots of negative numbers* Even roots: Negative numbers have no even roots. (undefined) Odd Roots: Negative numbers have negative roots.

Examples: Roots (of powers of 2) Even Roots: Odd Roots:

Roots: Number and Types Even Roots Odd Roots Positive 2 (one positive, one negative) 1 (positive) Negative 0 (undefined) 1 (negative)

Even Roots (of variable expressions)* When evaluating even roots (n is even) use absolute values (if resulting power is odd).

Odd Roots (of variable expressions)* When evaluating odd roots (n is odd) do not use absolute values.

Evaluating Roots of Monomials To evaluate nth roots of monomials: (where c is the coefficient, and x, y and z are variable expressions) or Simplify coefficients (if possible) For variables, evaluate each variable separately

Evaluating Roots of Monomials* To find a root of a monomial Split the monomial into a product of the factors, and evaluate the root of each factor. Variables: divide the power by the root Coefficients: re-write the number as a product of prime numbers with powers, then divide the powers by the root.

Use a calculator to find the roots* Find the following roots using a calculator (round to the 3 decimal places):

Examples: Simplify

Examples: Simplify