Algebra II 6.1-6.5

Property nth roots of nth powers

For any real number a, ^n√a^n=a or |a|

Plug back in the different values for x and the ones that work are the answers

How do you check for extraneous solutions

Like any other binomial

How do you multiply binomial radicals

Multiply the top and bottom by a radical with the same index as that of the denominator and add exponents to the terms in the radical that would get it to easily solve based off the index of the radical

How do you rationalize the denominator

Raise each side of the equation to the reciprocal of the exponent, n/m

How do you solve an equation in the form of x^m/n=k

Get rid of one radical at a time usually the one with the largest radicand first by squaring both sides of the equation

How do you solve an equation with multiple radicals

The nth root

If a^n=b with a and b real numbers and n a positive integer then a is an nth root of b

lxl

If m is even in the equation (x^m/n)^n/m, what does it equal

0

If m is negative in these what can a not equal: a^m/n=^n√a^m=(^n√a)^m

Postive and negative (2)

If the index is even and the radicand is positive there are how many real nth roots and what are they

1

If the index is odd how many real nth roots are there

^3/(12y^2z)^2 or ^3/12^2yz^2

If this was in the denominator ³√12y²z what are the two ways you could rationalize it

a^2b^3

Simplify this: ^3√a^6b^9

Principal root

The positive root of a radical with an even index and a positive radicand

^n/a^m=(^n/a)^m

What are the two radical forms of a^m/n let / in your answer be a radical sign

a^m/b^m

What does (a/b)^m equal

a^mn

What does (a^m)^n equal

a^mb^m

What does (ab)^m equal

^n/ab

What does ^n√a ×^n√b equal if ^n√a and ^n√b are real numbers and b is not equal to 0

^n/(a/b)

What does ^n√a/^n√b equal if ^n√a and ^n√b are real numbers and b is not equal to 0

a^m+n

What does a number with the same base and different exponent times a number with the same base and different exponent equal a^m × a^n

1/a^m

What does a^-m equal

a^m-n

What does a^m/a^n equal

x^1/n

What is the exponent form of ^n√x

Beginning

When determining if you should use an absolute value or not what part of the equation do you look at?

Index even, there is a variable under the radical, it starts with all even exponents and an answer with an odd exponent

When do you use absolute values around your answer

When the denominator is something plus or minus a radical

When do you use conjugates

When they have the same radicand and index

When is the only time you can add/subtract radicals

No

When to use absolute value around answer: Index odd

Maybe

When to use absolute value around answer: Index even

No

When to use absolute value around answer: Index even and there is a number under radical

Maybe

When to use absolute value around answer: Index even and there is a variable under the radical

Maybe

When to use absolute value around answer: Index even, there is a variable under the radical and it starts with all even powers

No

When to use absolute value around answer: Index even, there is a variable under the radical and it starts with at least one odd exponent

No

When to use absolute value around answer: Index even, there is a variable under the radical, it starts with all even exponents and an answer with an even exponent

Yes add absolute value

When to use absolute value around answer: Index even, there is a variable under the radical, it starts with all even exponents and an answer with an odd exponent

Index

n in both radicals

Radicand

the number under the radical sign -25