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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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What is a square root of 25? How many square roots does 25 have?

Definition: X is a square root of a if X² = a.

Symbolically, is the principle square root of a. To symbolically
represent each square root of a, one must write and − . This
leads to the short‐hand way of writing both square roots as ±.

Do the following square roots exist?

What are the following square roots?

In general….
is called a “radical sign” or a “root sign”.

A square root is a particular type of root that uses the root sign for
itself.

is an example of a radical expression since it an expression
with a root sign.

In the above expression, the 64z^4 is the radicand. The radicand is
the expression under (or better said, inside) a radical expression.

f (x) = is an example of a radical function.

Definition: A number S is called a perfect square if it’s the result of
squaring an integer.

You need to memorize the first 21 numeric perfect squares.

Variable expressions can be perfect squares
also if we amend the definition as follows:

An expression is a perfect square if its
coefficient satisfies the definition of a
numeric perfect square & each variable has
an integer exponent that is a multiple of 2.

The square root of a numeric value that isn’t a perfect square
usually results in an irrational number.

Recall that irrational numbers cannot be expressed as fractions of
integers and their decimal form neither repeats nor terminates.

Definition: X is a cube root of a if X³ = a.

All numbers have one cube root thus every cube root is a principle
cube root.

Definition: A number C is called a perfect cube if it’s the result of
cubing an integer.

You need to memorize the first 11 numeric perfect cubes.

Variable expressions can be perfect cubes
also if we amend the definition as follows:

An expression is a perfect cube if its
coefficient satisfies the definition of a
numeric perfect cube & each variable has
an integer exponent that is a multiple of 3.

Definitions:

X is a fourth root of a if X4 = a.
X is a fifth root of a if X5 = a.
X is an nth root of a if Xn = a.

All roots have an index. The index of a root is equal to the power
needed to return X to a by the previously state definitions.

Roots with an even index (such as square roots and fourth roots)…
Positive number have 2 real roots.
Zero is its own root.
Negative numbers have 0 real roots.

Roots with an odd index (such as cube roots and fifth roots)…
All numbers have exactly one real root.

Notationally write the fourth roots of 81 and evaluate.

Notationally write the fifth root of 243 and evaluate.

Definitions: A number R is called a perfect fourth if it’s the result of
raising an integer to a fourth power.
A number R is called a perfect fifth if it’s the result of
raising an integer to a fifth power.

 Perfect fourths: 0 1 16 81 256 625 Perfect fifths: 0 1 32 243 1024

For roots with even indices, keep in mind the following rule:

If variables can represent any real number, you may need to use absolute
value symbols when simplifying.

If the variables can only represent non‐negative numbers, you won’t need
absolute value symbols when simplifying.

Absolute value symbols are never needed if a root has an odd index.

Find each root. Assume that all variables represent non‐negative
real numbers.

Find each root. Assume that all variables can represent any real
number.

## §10.2 Radicals and Rational Exponents

Recall the Laws of Exponents (x ≥ 0)

Also:

Think about how the Laws of Exponents are related here:

Explore the possibilities associated with the following rational
exponent:

Thus in general we can conclude that

Evaluate and/or simplify.

Rewrite each expression in radical notation and simplify as possible.

Recall that which we can extend to define

Evaluate.

Use the properties of exponents to simplify each expression.

Multiply.

Factor.

Use rational exponents with each to find a single simplified radical.

Use rational exponents with each to find a single simplified radical.

## §10.3 Product & Quotient Rules and Simplifying Radical Expressions

Multiply.

Divide.

To simplify radicals, apply the product and quotient rules in reverse.
Simplify.

Simplify.

Simplify .

Emphasis: its all about perfect squares, cubes, etc.
Simplify.

Simplify.

Simplify.

Simplify.

Compare the following pairs of sums

Subtract.