Math Lesson 7.2.3 - Properties of Roots

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Welcome to our Math lesson on Properties of Roots, this is the third lesson of our suite of math lessons covering the topic of Roots, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

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:

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

we obtain

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,

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

and

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:

Indeed,

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,

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

and

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.

Indeed,

For example,

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

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

Indeed,

For example,

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

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

and

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

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 )
   = x^6/2 ∙ x^4/4 ∙ y^8/4/4 ∙ x^8/2 ∙ 3 ∙ y^4/2
   = 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.

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

Proof:

For example,

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

Indeed,

For example,

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

Indeed,

For example,

Example 6

Simplify the following expressions as much as possible.

1. √x ∙ y ∙ √x² ∙ y²
2. √ √x²⁰ - √√√x²⁷

Solution 6

1. Using the properties of roots, we obtain
   √x ∙ y ∙ √x² ∙ y²
   = √x ∙ y ∙ x² ∙ y²
   = √x³ ∙ y³)
   = √x³ ∙ √y³
   = x ∙ y
2. Again, using the properties of roots, we obtain
   √√x²⁰ - √√√x²⁷
   = √x^20/4 - √√x^27/3
   = √x⁵ - √√x⁹
   = √x⁵ - √x^9/3
   = √x⁵ - √x³
   = x - x
   = 0

More Roots Lessons and Learning Resources

Whats next?

Enjoy the "Properties of Roots" math lesson? People who liked the "Roots lesson found the following resources useful:

1. Properties Feedback. Helps other - Leave a rating for this properties (see below)
2. Powers and Roots Math tutorial: Roots. Read the Roots math tutorial and build your math knowledge of Powers and Roots
3. Powers and Roots Video tutorial: Roots. Watch or listen to the Roots video tutorial, a useful way to help you revise when travelling to and from school/college
4. Powers and Roots Revision Notes: Roots. Print the notes so you can revise the key points covered in the math tutorial for Roots
5. Powers and Roots Practice Questions: Roots. Test and improve your knowledge of Roots with example questins and answers
6. Check your calculations for Powers and Roots questions with our excellent Powers and Roots calculators which contain full equations and calculations clearly displayed line by line. See the Powers and Roots Calculators by iCalculator™ below.
7. Continuing learning powers and roots - read our next math tutorial: Standard Form

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