Chapter 10: Radical Expressions and Functions

Radical sign

Indicates a square root

Interpreting rational exponents

Exponents: 1/2 = integer squared, 1/3 = integer cubed, etc. The denominator of a fraction equals the index of a radicand. The numerator equals the exponent of the radicand.

Quotient rule

If you have a fraction as a radicand, it is the same as if you split it apart into a fraction with a root in the numerator and a root in the denominator. The opposite is also true, you can combine two roots in a fraction into one.

Power rule for exponents

Multiply the exponent outside of the parentheses with the exponent inside the parentheses.

Rationalizing the denominator having two terms

Multiply the numerator and denominator by the conjugate of the denominator.

Rationalize a denominator having one term

Multiply the numerator and denominator by the radical expression in the denominator.

Complex conjugates

Only a play conjugate rule to middle term.

Like radicals

Radicals that have the same index in the same radicand.

i2

Replace with -1

Simplifying expressions to absolute value

The exponent of a radicand can be canceled out by the index of the square root, simply leaving the radicand, less the exponent, as the square root and the absolute value.

Radicand

The expression under the radical sign

Nth Root

The index is greater than cubed.

Principal nth root

The positive root.

Principal square root

The positive square root

Adding and subtracting radical expressions

The radical expressions must be like terms.

Finding the domain of a square root function:

The radicand cannot be negative. Write an inequality that sets x equal to or greater than zero. Solve for x. Write answer in interval notation. That is the domain. If when you solve for X you get a negative number, the answer is all real numbers, because when looking for what X cannot be, a negative number will never occur.

Not a real number

The square root of a negative number.

Interpreting negative rational exponents

Turn the expression into a fraction, with the now positive radical notation on the bottom.

Product rule for radicands

You can multiply do radicals if they have the same index. The product of two nth roots is the nth root of the product of the radicands.

Standard form for a complex number

a+bi a = real number b= imaginary

The square root function

f(x)=sq root(x)

Powers of i

i0=1 i1= i i2=-1 i3=-i Pattern repeats for every 4 numbers. i to the remainder when n is divided by 4. n/4 i20/4 - no remainder=i0 =1

Square root function

Defined only for nonnegative inputs. Only in quadrant one of a graph.

Every positive number has:

Do square roots: one positive and one negative.

Rationalize a denominator

Do this to remove a radical expression from the denominator.

Perfect square 2-13

2=4 3=9 4=16 5=25 6=36 7=49 8=64 9=81 10=100 11=121 12=144 13=169

Perfect cube 2-6

2=8 3=27 4=64 5=125 6=216

Expression

Does not contain an =

To simplify a radical:

1. Determine the largest perfect nth power factor of the radicand. 2. Use the product rule to factor out and simplify this perfect nth power.

Steps to solving equations involving radical expressions

1. Isolate the radical on one side of the equation 2. Eliminate the square root, raise each side of the equation to the power equal to the index. 3. Solve 4. Check

To rationalize it denominator that is not a square

1. Write as an exponent. 2. Multiply by what is needed to make the fractional exponent whole.

Perfect power factor

A factor of a number that is also a perfect nth power. The nth power is designated by the index.

The cube root of a negative number

A negative real number.

Imaginary unit, i

A way to represent roots when there is a negative in an even root.

Pythagorean theorem

A2+B2=C2

Equation

Always contains an =. A statement that to expressions are equal.

Radical expression

An expression containing a radical sign

Power rule for products

Apply the exponent to each term within the parentheses.

To evaluate the root of a fraction

Apply the index to the numerator and the denominator.

Multiplying radicals with different indexes

Convert radicals to exponential form. Add them together and convert back to radical form.

Cubed root function

Defined for all inputs. Can be negative or positive.