Roots

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Powers and Roots Learning Material
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7.2Roots

In this Math tutorial, you will learn:

  • What are roots?
  • What is the geometrical meaning of square and cube roots?
  • What is the closest operation to roots?
  • Why roots are called fractional powers?
  • What is the absolute value of a number?
  • What are the properties of roots? Where do they derive from?
  • How to calculate the square root of a number manually?

Introduction

Can you find a number that, when multiplies by itself, gives 9? How did you find it/them? How many such numbers exist?

What operation is the above situation is related to? What is the inverse operation of it?

In this tutorial, we will deal with roots - the inverse of powers. In this way, we complete the framework of operations used in arithmetic or algebraic expressions.

The Meaning of Roots. Square and Cube Root

So far, we have learned far that the four basic arithmetic operations (addition, subtraction, multiplication and division) are related to each other in such a way that they are the inverse of each other two by two. In simpler words, subtraction is the inverse of addition and division is the inverse of multiplication.

When dealing with arithmetic or algebraic expressions, we have also dealt with powers, as a shorter way to calculate recurrent multiplications of the same factor by itself. Hence, logically, there must be an inverse operation of raise in power as well. And the truth is that such inverse operation does exist and is called finding the root of a number.

By definition, the nth root of a number is a new number that when multiplied n times by itself gives the original number.

The symbol we use to represent roots is (). The nth root of a number x is symbolically written as x. As a special case of roots, we can mention the 2nd root of a number, known as the "square root". In the case of square roots, the index n = 2 is not written but implied, just like the 'plus' symbol before positive numbers.

The name 'square root' originates from geometry, where the area of a square is calculating by raising the side length in the second power. Therefore, when the area of a square is known, we calculate the side length by taking the square root of the value representing the area, as shown in the figure below.

Math Tutorials: Roots Example

(The symbol ( => ) means 'implies' in math.)

A positive number has two square roots: one positive and another negative. Thus, for example we write

√16 = ± 4

because

( + 4)2 = ( + 4) ∙ ( + 4) = ( + 16)

and

(-4)2 = (-4) ∙ (-4) = ( + 16)

Example 1

Calculate the square root of the following numbers:

  1. 49
  2. 81
  3. 196
  4. 361

Solution 1

  1. We have
    √49 = ± 7 because (-7)2 = (-7) ∙ (-7) = 49 and ( + 7)2 = ( + 7) ∙ ( + 7) = 49
  2. We have
    √81 = ± 9 because (-9)2 = (-9) ∙ (-9) = 81 and ( + 9)2 = ( + 9) ∙ ( + 9) = 81
  3. We have
    √196 = ± 14 because (-14)2 = (-14) ∙ (-14) = 196 and ( + 14)2 = ( + 14) ∙ ( + 14) = 196
  4. We have
    √361 = ± 19 because (-19)2 = (-19) ∙ (-19) = 361 and ( + 19)2 = ( + 19) ∙ ( + 19) = 361

Obviously, the square root of negative numbers does not exist in the set of real numbers, as we cannot find any negative number which, when raised at the second power, gives a positive value. Not only is this is true for square root but for all even roots, as we can pair them two by two.

Another root that has a geometry-based meaning is the "cube root" of a number. It represents a new number which when raised to the third power gives the original number. We write the index 3 at the upper left part of the root symbol to highlight the fact that we are dealing with the third root (or cube root) of a number.

The term 'cube root' derives from the fact that the volume of a cube is calculated by raising its side length to the third power. Therefore, if the volume is known, the side length of a cube is obtained by calculating the cube root of volume.

Math Tutorials: Roots Example

For example,

∛8 = 2 because 23 = 8
∛125 = 5 because 53 = 125
∛(-64) = -4 because (-4)3 = -64

etc.

From the above examples, it is easy to notice two things:

  1. The cubic root of a number is a single value, not two numbers with opposite signs as in square root.
  2. Unlike in square root, the cubic root of negative numbers does exist and it is negative.

Example 2

Calculate the following roots.

  1. -1
  2. 27
  3. -216

Solution 2

  1. We have
    ∛(-1) = -1 because (-1)3 = (-1) ∙ (-1) ∙ (-1) = -1
  2. We have
    ∛27 = 3 because 33 = 3 ∙ 3 ∙ 3 = 27
  3. We have
    ∛(-216) = -6 because (-6)3 = (-6) ∙ (-6) ∙ (-6) = -216

Roots as Fractional Powers

From the definition of roots, we have

x = y ⟹x = yn

This means we can write x as

x = y ∙ y ∙ y…n times

Hence, we can express y as

y = x1/n

because multiplying n times both terms above, yields

y ∙ y ∙ y…n times = x1/n ∙ x1/n ∙ x1/nn times = x1/n + 1/n +1/nn times = xn/n = x1 = x

Therefore, the two expressions

x and x1/n

are equivalent (they show the same thing).

In this way, it is easy to conclude that roots are nothing more but simply fractional powers.

Example 3

Calculate the following values:

  1. 251/2
  2. 641/3
  3. (-32)1/5

Solution 3

From the roots - fractional powers equivalence, we have

  1. 251/2 = √25 = ± 5 because (-5)2 = 25 and ( + 5)2 = 25
  2. 641/3 = ∛64 = 4 because 43 = 4 ∙ 4 ∙ 4 = 64
  3. -321/5 = -32 = -2 because (-2)5 = (-2) ∙ (-2) ∙ (-2) ∙ (-2) ∙ (-2) = -32

Properties of Roots

As an alternative form of representing fractional powers, roots have more or less the same properties of powers. Henceforth, we will consider, as default, only the positive value of the root unless this is not explicitly stressed. Let's explain all of them.

Property 1: For any real non-negative numbers a and b, the following property is true:

√a ∙ √b = √(a ∙ b)

Indeed, using the root-fractional power equivalence and given the fourth property of indices

ac× bc= (a × b)c

we obtain

√a ∙ √b = a1/2 ∙ b1/2 = (a ∙ b)1/2 = √(a ∙ b)

This property is particularly useful when we multiply two numbers which don't have their square root a whole number, but after multiplication, the product has a whole root.

For example,

√2 ∙ √18 = √(2 ∙ 18) = √36 = ± 6

If we tried to calculate the square roots separately, we would obtain two irrational numbers, namely

√2 = 1.414213562373…

and

√18 = 4.242640687119…

which we would be compelled to round up when multiplying, decreasing therefore the accuracy of result.

Property 2: For any real non-negative number a and positive real number b, the following property is true:

√a/√b = √(a/b)

Indeed,

√a/√b = a1/2/b1/2 = (a/b)1/2 = √(a/b)

This is because the fourth property of indices is true not only when two different factors are raised to the same power but also when two numbers that are related to each other through division are raised to the same power.

For example,

√48/√3 = √(48/3) = √16 = 4

Again, if we tried to calculate the square roots separately, we would obtain two irrational numbers, namely

√48 = 6.928203230…

and

√3 = 1.732050807…

Again, doing the operations using the above numbers, we would obtain an irrational number that is close to the exact value (4) but not precisely equal because of rounding.

Example 4

Calculate the value of the following expressions.

  1. √14 ∙ √7/√2
  2. √5 ∙ √15/√3

Solution 4

  1. Combining the first and the second properties of roots, yields
    √14 ∙ √7/√2 = √(14 ∙ 7)/√2
    = √98/√2
    = √(98/2)
    = √49
    = 7
  2. Combining the first and the second properties of roots, yields
    (√5 ∙ √15/√3 = √(5 ∙ 15)/√3
    = √75/√3
    = √(75/3)
    = √25
    = 5

Property 3: For any real numbers a and n, the following property is true.

(√a)n = √(an )

Indeed,

(√a)n = √a ∙ √a ∙ √a…n times = √(a ∙ a ∙ a ∙ …) n times = √(an )

For example,

√(163 ) = (√16)3 = 43 = 64

This property is very useful, as it avoids dealing with big numbers. Thus, if the above property were not applied, we would obtain

√(163 ) = √(16 ∙ 16 ∙ 16)
= √4,096
= 64

Property 4: For any real numbers a, b and for any positive integer n ≥ 2, the following property is true.

ab = ab/n

Indeed,

ab = a ∙ a ∙ a ∙ …b times
= aaa ∙ …b times
= a1/n ∙ a1/n ∙ a1/n ∙ …b times
= a1/n + 1/n + 1/n + …b times
= ab/n

For example,

∛(26 ) = 26/3 = 22 = 4

Indeed, if the above property were not used, we would obtain

∛(26 ) = ∛(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2) = ∛64 = 4 because 43 = 64

Property 5: For any number a and any integer n ≥ 2, the following property is true.

an = a if n is odd

and

an = |a| if n is even

The symbols (||) indicate the absolute value of a number that is its distance from the origin regardless the direction in the number axis. For example, |-5| = 5 because the number -5 is five units away from the origin (i.e. from zero). Likewise, | + 5| = 5 as well, because the distance of + 5 from the origin is also five units.

The fifth property of roots derives from the previous one (fourth property). It represents a special case where b = n. However, given the convenience it creates during the operations, we wanted to highlight it by considering this part of the fourth property as a separate one.

For example, if we have to calculate ∛8, we can write

∛8 = ∛(23 ) = 2

instead of trying to figure out the third root of 8.

Example 5

Calculate the value of the following expressions.

  1. ∛27 ∙ √9 ∙ ∜16/√64 ∙ ∜81
  2. √(x6 ) ∙ ∜(x4 ∙ y8 )/√(16x8 ) ∙ √(9y4 )

Solution 5

  1. Using the fifth property of roots, yields
    ∛27 ∙ √9 ∙ ∜16/√64 ∙ ∜81
    = ∛27 ∙ √9 ∙ ∜16/√64 ∙ ∜81
    = ∛(33 ) ∙ √(32 ) ∙ ∜(24 )/√(82 ) ∙ ∜(34 )
    = 3 ∙ 3 ∙ 2/8 ∙ 3
    = 3 ∙ 2/8
    = 6/8
    = 3/4
  2. Using the properties of roots, yields
    √(x6 ) ∙ ∜(x4 ∙ y8 )/√(16x8 ) ∙ √(9y4 )
    = √(x6 ) ∙ ∜(x4 ) ∙ ∜(y8 )/√16 ∙ √(x8 ) ∙ √9 ∙ √(y4 )
    = x6/2 ∙ x4/4 ∙ y8/4/4 ∙ x8/2 ∙ 3 ∙ y4/2
    = x3 ∙ x ∙ y2/12 ∙ x4 ∙ y2
    = x4 ∙ y2/12 ∙ x4 ∙ y2
    = 1/12

Property 6: For any real a and b and for any positive integer n, the following property is true.

a ∙ b = √(an ∙ b)

This property is very useful when trying to leave inside the root the least possible variables or numbers.

Proof:

a ∙ b = a1 ∙ b1/n = n/n ∙ b1/n = anb = √(n&an ∙ b)

For example,

∛40 = ∛(8 ∙ 5)
= ∛8 ∙ ∛5
= ∛(23 ) ∙ ∛5
= 2 ∙ ∛5

Property 7: For any real number a and b for any integers m and n, the following property is true.

a = am

Indeed,

am = am/n ∙ m = a1/n = a

For example,

√(6&343) = am = √(2&7) = √7

Property 8: For any real number a for any integers m and n, the following property is true.

a = a

Indeed,

a= (a)1/m = (a1/n )1/m = a1/n1/m = a1/n ∙ m = a

For example,

64 = 64 = 64 = 2

Example 6

Simplify the following expressions as much as possible.

  1. x ∙ yx2 ∙ y2
  2. x20 - x27

Solution 6

  1. Using the properties of roots, we obtain
    x ∙ yx2 ∙ y2
    = x ∙ y ∙ x2 ∙ y2
    = x3 ∙ y3)
    = x3y3
    = x ∙ y
  2. Again, using the properties of roots, we obtain
    x20 - x27
    = x20/4 - x27/3
    = x5 - x9
    = x5 - x9/3
    = x5 - x3
    = x - x
    = 0

How to Manually Calculate the Square Root of a Number?

Despite the fact that nowadays we are in the age of technology and almost every operation is carried out with computers or calculators, it is important to understand the traditional method of calculating the square root of a number, similarly to other operations we have explained in the first chapter of this course.

At first glance, the method used to manually calculate the square root of a number looks very similar to division by hand already explained in the first chapter. Let's explain how to find the square root step by step. In this paragraph, we are going to consider only perfect squares, i.e. numbers that have their square root a natural number. For example, let's take the number 729 for illustration (the calculator gives √729 = 27).

Step 1: Split the original number into pairs from right to left. The first digit may end up alone but this is OK.

Math Tutorials: Roots Example

Step 2: Take the closer (smaller) square root of the leftmost chunk (here, the closest square root of 7 is 2) and write it on a separate section, similar to the one used to write the quotient in the division.

Math Tutorials: Roots Example

Step 3: The number obtained is squared and the result is written below the original chunk considered.

Math Tutorials: Roots Example

Step 4: Then, we subtract the new number from the original chunk, like in division.

Math Tutorials: Roots Example

Step 5: Now, we write the other pair next to the remainder. Another thing that we do in this step is to double the number found at the beginning (2 × 2 = 4), as this number will be used in the next step. Since here 2 × 2 corresponds to 22 - an operation we did in the third step - we have not written this operation (2 × 2) but normally, it must be done.

Math Tutorials: Roots Example

Step 6: Now, we must find a suitable digit which in this case must be the unit digit of the two numbers in the operation 4 _ × _ = 329. In our case, the number required is 7 because 47 × 7 = 329. This number is written next to 2 in the upper right section. The complete outline of the operation is shown below, where all steps are highlighted in different colours.

Math Tutorials: Roots Example

Example 7

Manually calculate the following square roots.

  1. √2,809
  2. √11,236

Solution 7

  1. The complete outline of the operation is shown below, where all steps are highlighted in different colours. Math Tutorials: Roots Example Thus,
    √2,809 = 53
  2. Again, the complete outline of the operation is shown below, where all steps are highlighted in different colours. Math Tutorials: Roots Example Thus,
    √11,236 = 106

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