Algebra Tutorials!  
Wednesday 24th of July
Square Roots and Radical Expressions
Solving Radical Equations
Simplifying Radical Expressions
Irrational Numbers in General and Square Roots in Particular
Roots of Polynomials
Simplifying Radical Expressions
Exponents and Radicals
Products and Quotients Involving Radicals
Roots of Quadratic Equations
Radical Expressions
Radicals and Rational Exponents
Find Square Roots and Compare Real Numbers
Radicals and Rational Exponents
Theorems on the Roots of Polynomial Equations
Simplifying Radical Expressions
Exponents and Radicals
Properties of Exponents and Square Roots
Solving Radical Equations
Rational Exponents and Radicals,Rationalizing Denominators
Rational Exponents and Radicals,Rationalizing Denominators
Quadratic Roots
Exponents and Roots
Multiplying Radical Expressions
Exponents and Radicals
Solving Radical Equations
Solving Quadratic Equations by Factoring and Extracting Roots
Newton's Method for Finding Roots
Roots of Quadratic Equations Studio
Roots, Radicals, and Root Functions
Review division factoring and Root Finding
Simplifying Radical Expressions
Multiplying and Simplifying Radical Expressions
Multiplication and Division of Radicals
Radical Equations
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Exponents and Radicals

Solutions to Examples

2.1. Solution:

Prop. Rads. #3
Prop. Rads. #1
A Prop. Abs. Value #


Example Notes: In equation (S-1), no absolute values were needed for
the x4 since x4 ≥ 0; however, it is possible for y5 to be negative, hence
the absolute values were needed.
Example 2.1.

2.2. Solution:

defn of a-n
Prop. Rads. #3
Law Exp. #3
Prop. Rads. #1

Example 2.2.

Important Points

That’s Right! The correct answer is (b).
The square of −3 = (−3)2 = (−3)(−3) = 9

First choice or second? If you chose (a) as your first choice, you have
a weakness in this area. Some students erroneously write −32 when
what they really mean is (−3)2. A mathematician would interpret the
expression −32 as −(3)(3) = −9; that is the correct interpretation of
the notation −32 is the negative of the number3 squared.

You must be careful about writing −32 when, in reality, you mean
(−3)2. You and the one grading your paper might have a difference
in opinion about the meaning of the symbol—guess who wins the

Therefore, when raising a number that has a negative sign, always
enclose the number, including negative sign, with parentheses. Thus,
cube of the number −2 is (−2)3 = (−2)(−2)(−2) = −8.

Similar comments can be made about symbolic numbers: The cube of
the number −x is (−x)3 not −x3. More on this later.
Important Point

Good Choice! The correct response is ‘n.o.t.’, which means ‘none of

A common error students make when (trying) to manipulate radicals
is to essentially write (on a test paper, for example)

But this is an algebraic blunder! Don’t do that! The root of a sum of
two quantities in not equal to the sum of the roots. A simple example
will illustrate

Symbolic quantities such as a and b represent numbers. Equations you
write involving symbolic quantities must be true when the symbols
are replaced with numbers. To see that the root of a sum is not equal
to the sum of the roots, just give a and b appropriate values.

Take a = 9 and b = 16. Thus,

Look at the extreme left and right sides of this nonequation. What
do you see?

That is,

Don’t make this mistake ever again! Important Point

Correct Again! It is not always true that Sometimes the
radical equals a and sometimes it doesn’t equal a. Here are some

Example 1. Situation where is false. Think of a = −1 and
n = 2, then

Example 2. Situation where is true. Think of a = 1 and
n = 2, then

Sometimes it true, sometimes its false. When trying to simplifying a
symbolic expression like it turns out that we need to know the
sign of the number a.

Continue now the discussion following this quiz for a definitive explanation
of how to simplify radicals like . Important Point

Way to go, mate! Because we are taking the square root of x2y, it is
an implicit assumption that x2y ≥ 0. Since x2 ≥ 0 regardless of the
value of x, we deduce y ≥ 0.

Now by Law #2 of the Properties of Radicals we have

Important Point

The original expression was from which we deduce that x3 ≥ 0;
hence, x ≥ 0. Thus, |x| = x since x ≥ 0. ’Nuff Said!
Important Point


The last step needs some comment. Because we started with the expression
we conclude that implies that x ≥ 0,
because we are dealing with an odd power of x.

Finally, x ≥ 0 implies |x| = x. What was simple!

Awareness of the signs of the quantities is often essential to a successful
simplification. Important Point

Way to go! Convert to exponential notation, reduce fractions, then
return to radical notation.

Important Point

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