Algebra-2 Section 7-4 Radicals with Index Numbers Higher than 2

Quiz 7-3 Simplify the following:

Vocabulary Conjugates: (x + 2)(x – 2) When you multiply any binomial by its conjugate, the “middle term” of the trinomial “cancels out”. When you multiply a binomial that has a radical term, by its conjugate, the “middle term” of the trinomial “cancels out”  the radical term disappears. the radical term disappears.

4 Your turn Simplify: (multiply the binomials) 1. 2.

5 “Rationalize” the Denominator Irrational numbers NOT ALLOWED In the denominator Multiply top & bottom by the conjugate

6 Your turn: 3. Rationalize the denominator: 4. Rationalize the denominator:

4-57 Square Root Index Radical Radicand

Vocabulary Index of the radical: the number of times a number must be used as a factor (of the radicand) in order to “bring it outside” of the radical. Radicand Index What number of factors of factors am I looking for? am I looking for?

Your Turn: Simplify

Vocabulary: 3 rd Root of 5 : 3 rd Root of 8: Index

Your turn: Write the following in “radical form” 5 th Root of Write the following in “English”

Radicands containing variables Why is this true? Index number  what number, that is under the radical, is used as a factor (index number) times? ‘x’ is used as a factor 5 times.

Radicands containing variables Using the Product Property of exponents, factor the radicand into terms with the same exponent as the index number of the radical. Simplifies to:

Your turn: simplify

Radicands containing many variables Using the Product Property of exponents, factor the radicand into terms with the same exponent as the index number of the radical.

Radicands containing many variables

Your turn: simplify

Ratios of Radicals Split the radical of a quotient into the quotient of radicals. into the quotient of radicals.

Ratios of Radicals If you don’t see the index number what is it? This not so scary. How would you simplify the radicand? Put radicand back inside the radical, then simplify the radical

Ratios of Radicals This not so scary. How would you simplify the radicand? Put radicand back inside the radical, then simplify the radical

Your turn: simplify the following

Even and Odd roots of -1 What number used as a factor 2 times equals -1?

Even and Odd roots of -1 What number used as a factor 3 times equals -1?

Even index numbers and negative radicands What number used as a factor 2 times equals -8?

Even index numbers and negative radicands What number used as a factor 4 times equals -16?

Odd index numbers and negative radicands What number used as a factor 3 times equals -8?

Odd index numbers and negative radicands What number used as a factor 3 times equals -125?

Your turn: Simplify 2i2i2i2i Even index number and negative radicand and negative radicand If the index is odd there is a real solution there is a real solution if the radicand is negative. if the radicand is negative

Negative Radicand summary: Even index number: imaginary number Odd index number: The solution is negative only.