1. The document provides an overview of roots and radicals, including definitions of square roots, principal and negative square roots, radical expressions, rational and irrational numbers, and methods for simplifying, adding, subtracting, multiplying, dividing, and rationalizing radicals. 2. Square roots are factors whose product is the radicand, and the principal square root is the positive root. Radicals indicate roots and have a radicand and index. Rational numbers can be expressed as ratios while irrational numbers cannot. 3. Graphs and tables can be used to find approximate square roots. Radicals can be simplified using properties of exponents and fractions. Radical equations are solved by isolating the radical and taking squares of both

1. CHAPTER 12
ROOTS AND RADICALS
Presented In The Course of Elementary Algebra
1 𝑡ℎ
Semester in Academic Years 2017/2018
By
1. MeutiahNahrisyah/1711441014
2. NiatiIndan/1711441013
Program of ICP Mathematics Education
Mathematics Department
UNIVERSITAS NEGERI MAKASSAR

2. “ROOTS AND RADICALS”
1. UNDERSTANDING ROOTS AND RADICALS
A square root of a number is one of two equal factors of the number.
For every pair of realnumbers a and b, if 𝑎2 = 𝑏, then a is called a square root of
b.
Thus, +5 is a square root of 25 since (+5) (+5) = 25
Also -5 is another square root of 25 since (-5) (-5) = 25
RULE :
A positive number has two square roots which are opposites of each other (same
absolute value but unlike signs).
Thus, +10 and −10 are the two square roots of 100.
To indicate both square roots, the symbol ±, which combines + and -, may be
used.
Thus, the square root of 49, +7 and -7, may be written together as ±7.
Read ±7 as “plus or minus 7”.
The principal squre root of a number its is positive square root.
Thus, the principal square root of 36 is +6
The symbol √ indicates the principal square root of a number.
Thus, √9 = principal square root of 9 = +3
To indicate the negative square root of a number, place the minus sign before √.
Thus, −√16 = −4.
NOTE :
Unless otherwise stated, whenever a square root of a number is to be found, it is
understood that the principal or positive square root is required.

3. Parts of a Radical Expression
1. A radical is an indicated root of a number or an expression.
Thus, √5, √8𝑥3
, and ∜7𝑥3 are radicals.
2. The symbol √ , √
3
, and √
4
are radical signs.
3. The radicans is the number or expression under the radical sign.
Thus, 80 is the radicand of ∛80 or ∜80.
4. The index of a root is the small number written above and to the left of the
radical sign √ .The index indicates which root is to be taken. In square root,
the index 2 is not indicated but is understood.
Thus, ∛8indicates the third root or cube of 8.
Nth Roots
Other roots can be found, as well
The nth root of a is defined as
√ 𝑎𝑛
= 𝑏 only if 𝑏 𝑛 = 𝑎
If the index, n, is even, the root is NOT a realnumber when 𝑎 is negative.
If the index is odd, the root will be a real number.
2. UNDERSTANDING RATIONAL AND IRRATIONAL NUMBERS
Rational Numbers
A rational numbers is one that can be expressed as quotient or ratio of two
integers.
Kind of Rational Numbers
1. All integers, that is, all positive and negative whole numbers and zero.
Thus, 5 is rational since 5 can be expressed as
5
1
.
2. Fraction, whose numerator and denominator are integers, after simplification.
Thus,
1.5
2
is rational since because it aquals
3
4
when simplified.
3. Decimals which have a limited number of decimal places.

4. Thus, 3.14 is rational since it can be expressed as
314
100
.
4. Decimals which an unlimited number of decimal places and the digits
continue to repeat themselves.
Thus, o.6666… is rational since it can be expressed as
2
3
.
Note: … is a symbol meaning “continued without end”.
5. Square root expressions whose radicand is a perfect square,such as √25;
cube root expressions whose radicand is a perfect cube,such as √273
; etc.
Irrational Numbers
An irrational number is one that cannot be expressed as the ratio of two intrgers.
Kind of irrational numbers
 √2 is an irrational number since it cannot equal a fraction whose terms are
integers.
 𝜋 is also irrational
 Other examples of irrationals are
√5
2
,
2
√5
and 2 + √7
.
3. FINDING THE SQUARE ROOT OF A NUMBER BY USING A
GRAPH
Approximate square roots of a numbers can be obtained by using a graph of 𝑥 =
√ 𝑦.
Table of values used to graph 𝑥 = √ 𝑦 from 𝑥 = 0to 𝑥 = 8
If 𝑥 =
0
1 1
1
2
2 2
1
2
3 3
1
2
4 4
1
2
5 5
1
2
6 6
1
2
7 7
1
2
8
If 𝑦 =
0
1 2
1
4
4 6
1
4
9 12
1
4
16 20
1
4
25 30
1
4
36 42
1
4
49 56
1
4
64

5. To find the square of a number graphically, proceed in the reserve direction;
from the 𝑥-axis to the curve, then to the 𝑦-axis.
4. FINDING THE SQUARE ROOT OF A NUMBER BY USING A
TABLE
Approximate square roots of numbers can be obtained using the table. The square
root values obtained from such a table more precise than those from the graph of
𝑥 = √ 𝑦.
[Using the table on page 290] To find the principal square root of a number, look
for the number under 𝑁. Read the square root of the number under √ 𝑁,
immediately to the right of the number.
5. SIMPLIFYING THE SQUARE ROOT OF A PRODUCT
Formulas: √ 𝑎𝑏 = √ 𝑎 ∙ √ 𝑏, √ 𝑎𝑏𝑐 = √ 𝑎 ∙ √ 𝑏 ∙ √ 𝑐
Ex: √1600 = √16 ∙ √100 = 4 ∙ 10 = 40
To simplify the square root ofa product
Simplify √72.

6. Procedure Solution
Factor the radicand, choosing perfect square factors: √(4)(9)(2)
Form a separate square root for each factor: √4√9√2
Extract the square roots of each perfect square: (2)(3)√2
Multiply the factors autside the radical: 6√2
To simplify the square root ofa powers
Keep the base and take one-half of the exponent
Thus, √𝑥6 = 𝑥3 since 𝑥3 ∙ 𝑥3 = 𝑥6
To simplify the square root ofthe product ofthe powers
Keep each base and take one-half of the exponent.
Thus, √𝑥2 𝑦4 = 𝑥𝑦2 since √𝑥2 𝑦4 = √𝑥2 ∙ √𝑦4 = 𝑥𝑦2
6. SIMPLIFYING THE SQUARE ROOT OF A FRACTION
√
𝑎
𝑏
=
√ 𝑎
√ 𝑏
The square root of a fraction equals the square root of the numerator divided by
the square root of the denominator
Ex,,√
16
25
=
√16
√25
=
4
5
To simplify the square root of a fraction whose denominator is not a perfect
square, change the fraction to an equivalent fraction which has a denominator
that is the smallest perfect square.
Ex, √
1
8
= √
2
16
=
√2
√16
=
√2
4
=
1
4
√2
7. ADDING AND SUBTRACTING SQUARE ROOT OF NUMBER
Like radical are radicals having the same index and the same radicand.

7. Ex,
 5√3 𝑎𝑛𝑑 2√3
 8√ 𝑥 𝑎𝑛𝑑 3√ 𝑥
 7√6 𝑎𝑛𝑑 5√6
To combine like radicals, keep the common radical and combine their
coefficients.
Ex,
3√2 + 2√2
= (3 + 2)√2
= 5√2.
Unlike radicals,may be combined into one radical if like radicals can be
obtained by simplifying.
Ex,
 5√3 𝑎𝑛𝑑 3√2
 8√ 𝑥 𝑎𝑛𝑑 3√ 𝑦
 7√63
𝑎𝑛𝑑 3√6
Combining unlike radical after simplification
Ex,
√5 − √20 + √45
= √5 − √4 × 5 + √9 × 5
= √5 − 2√5 + 3√5
= 2√5.
8. MULTIPLYING AND DIVIDING SQUARE ROOTS OF NUMBERS
Multiplying square roots
𝑥√ 𝑎 ∙ 𝑦√𝑏 = 𝑥𝑦√𝑎𝑏

8. 𝑥√ 𝑎 ∙ 𝑦√𝑏 ∙ 𝑧√ 𝑐 = 𝑥𝑦𝑧√𝑎𝑏𝑐
To multiply square root monomials
Multiply 3√3 × 4√2
Procedure Solution
Multiply coefficient and radicals separately (3 × 4)√3 × 2
Multiply the resulting product 12√6
Simplify, if possible 12√6
Dividing square roots
𝑥√ 𝑎
𝑦√ 𝑏
=
𝑥
𝑦
√
𝑎
𝑏
Assuming 𝑏 ≠ 0
To divide square root monomials
Divide
6√10
3√2
Procedure Solution
Divide coefficient and radicals separately
6√10
3√2
Divide the resulting product 2√5
Simplify, if possible 2√5
9. RATIONALIZING THE DENOMINATOR OF A FRACTION
To rationalize the denominator of a fraction is to change the denominator from an
irrational number to a rational number. This process involves multiplying the
quotient by a form of that will eliminate the radical in the denominator
Ex,
4
√2
=
4
√2
∙
√2
√2
=
4√2
2
= 2√2

9. Note: we need to multiply by the conjugate of the numerator or denominator. The
conjugate uses the same terms, but the opposite operation.
Ex,
√3+2
√2+√3
∙
√2−√3
√2−√3
=
√6−3+2√2−2√3
2−3
=
√6−3+2√2−2√3
−1
= −√6 − 3 + 2√2 − 2√3
10. SOLVING RADICAL EQUATION
Radical equation are equations in which the unknown is included in a radicand.
To solve a radical equation
Solve √2𝑥 + 5 = 9
Procedure Solution
Isolate the term containing the
radical
√2𝑥 + 5 = 9
√2𝑥 = 4
Square both sides
By squaring, 2𝑥 = 16
Solve for the unknown
𝑥 = 8
Check the roots obtained in the
original equation
Check for 𝑥 = 8
√2𝑥 + 5 = 9 ; √16 + 5 =
9 ; 9 = 9 (ans.)