iCalculator™ Math

Multiplication And Division Of Roots Calculator

The Multiplication And Division Of Roots Calculator will calculate:

The product and quotient of two roots.

Multiplication And Division Of Roots Calculator Parameters:

Roots must have positive integer indices with a value of 2 or more.
Numbers inside even roots must be positive.

Multiplication And Division Of Roots Calculator
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Calculator Precision (Decimal Places)
√x^a
The number in the first root (x)
Power of the number in the first root (a)
Index of the first root (m)
√y^b
The number in the second root (y)
Power of the number in the second root (b)
Index of the second root (n)

Multiplication And Division Of Roots Calculator Results (detailed calculations and formula below)
Product of two roots Formula and Calculations
Product of two roots, P =
Quotient of two roots, Q =
√x^a ∙ √y^b = √x^an ∙ y^bm √ ∙ √ = √^∙ ∙ ^∙ √ ∙ √ = √ ∙ √ ∙ √ = √ ∙ ∙ = √ = Note: their may be slight variance in the equating results due to the rounding effect of the calculator when processing each step of the formula and subsequent calculations
Quotient of two roots Formula and Calculations
√x^a/√y^b = √**x^an/y^bm √/√ = √^∙/^∙ √/√ = √/ √/√** = √/ / = √ = Note: their may be slight variance in the equating results due to the rounding effect of the calculator when processing each step of the formula and subsequent calculations
Multiplication And Division Of Roots Calculator Input Values
The number in the first root (x)
Power of the number in the first root (a)
Index of the first root (m)
The number in the second root (y)
Power of the number in the second root (b)
Index of the second root (n)

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each multiplication and division of roots calculation, the Multiplication And Division Of Roots Calculator will automatically calculate the results and update the formula elements with each element of the multiplication and division of roots calculation. You can then email or print this multiplication and division of roots calculation as required for later use.

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$Multiplication And Division Of Roots. This image shows the properties and multiplication and division of roots formula for the Multiplication And Division Of Roots$

Theoretical description

Roots represent another form of writing numbers in fractional (rational) power. This means the following expressions are equivalent

√x^m = x^m/n

If n = 2, this index does not appear in the root but it is implied. We call it the square root of the number. On the other hand, when the index n = 3, we are dealing with the cubic root of the given number.

For example,

√2⁶ = 2^6/3
= 2²
= 4

This result could have been obtained in a different way:

√2⁶ = √64 = 4

This is true because 4³ = 64.

The following rules are applied when multiplying and dividing two roots:

If the two roots have the same index (n₁ = n₂ = n), then

√x ∙ √y = √x ∙ y

and

√x/√y = √x/y

This is true because

√x ∙ √y = x^1/n ∙ y^1/n = (x ∙ y)^1/n = √x ∙ y

and

√x/√y = x^1/n/y^1/n = (x/y)^1/n = √x/y

For example,

√8 ∙ √27 = √8 ∙ 27 = √216 = 6 because 6³ = 216

Indeed,

√8 = 2 and √27 = 3

Thus,

√8 ∙ √27 = 2 ∙ 3 = 6

Another example,

√64/√8 = √64/8 = √8 = 2 because 2³ = 8

Indeed,

√64 = 4 and √8 = 2

Thus,

√64/√8 = 4/2 = 2

If the two roots have the same index (n₁ = n₂ = n), then

√x^a ∙ √y^b = √x^a ∙ y^b

and

√x^a/√y^b = √x^a/y^b

This is true because

√x^a ∙ √y^b = x^a/n ∙ y^b/n = (x^a ∙ y^b )^1/n = √x^a ∙ y^b

and

√x^a/√y^b = x^a/n/y^b/n = x^a/y^b^1/n = √x^a/y^b

For example, if n = 2, then

√2⁴ ∙ √4³ = √2⁴ ∙ 4³ = √16 ∙ 64 = √1024 = 32

Indeed,

√2⁴ = √16 = 4 and √4³ = √64 = 8

Thus,

√2⁴ ∙ √4³ = 4 ∙ 8 = 32

Another example,

√10⁶/√5⁴ = √10⁶/5⁴ = √1,000,000/625 = √1600 = 40

Indeed,

√10⁶ = √1,000,000 = 1000 and √5⁴ = √625 = 25

Thus,

√10⁶/√5⁴ = 1000/25 = 40

If the two roots have different indices m and n (m ≠ n), then

√x^a ∙ √y^b = √x^an ∙ y^bm

This is true because

√x^a ∙ √y^b = x^a/m ∙ y^b/n = x^an/mn ∙ y^bm/mn = (x^an ∙ y^bm )^1/mn = √x^an ∙ y^bm

and

√x^a/√y^b = √x^an/y^bm

This is true because

√x^a/√y^b = x^a/m/y^b/n = x^an/mn/y^bm/mn = (x^an/y^bm)^1/mn = √x^an/y^bm

For example,

√5⁶ ∙ √2⁸ = ^{3 ∙ 2}√5^{6 ∙ 2} ∙ 2^{8 ∙ 3}
= √5¹² ∙ 2²⁴
= √244,140,625 ∙ 16,777,216
= √4,096,000,000,000,000
= 400

Indeed,

√5⁶ = 5^6/3 = 5² = 25 and √(2⁸ ) = 2^8/2 = 2⁴ = 16

Thus,

√5⁶ ∙ √(2⁸ ) = 25 ∙ 16 = 400

Another example,

√4⁵/√100³ = ^{10 ∙ 6}√4^{5 ∙ 6}/100^{3 ∙ 10}
= √4³⁰/100³⁰
= √(4/100)³⁰
= 4^30/60/100^30/60
= 4^1/2/100^1/2
= √4/√100
= 2/10

Indeed,

√4⁵ = 4^5/10 = 4^1/2 = √4 = 2 and √100³ = 100^3/6 = 100^1/2 = √100 = 10

Thus,

√4⁵/√100³ = 2/10

Obviously, you don't have to make all these calculations. You just have to insert the numbers and the calculator will calculate the result automatically.

Powers and Roots Math Tutorials associated with the Multiplication And Division Of Roots Calculator

The following Math tutorials are provided within the Powers and Roots section of our Free Math Tutorials. Each Powers and Roots tutorial includes detailed Powers and Roots formula and example of how to calculate and resolve specific Powers and Roots questions and problems. At the end of each Powers and Roots tutorial you will find Powers and Roots revision questions with a hidden answer that reveal when clicked. This allows you to learn about Powers and Roots and test your knowledge of Math by answering the revision questions on Powers and Roots.

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