# Multiplication And Division Of Roots Calculator

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The Multiplication And Division Of Roots Calculator will calculate:

1. The product and quotient of two roots.

Multiplication And Division Of Roots Calculator Parameters:

1. Roots must have positive integer indices with a value of 2 or more.
2. Numbers inside even roots must be positive.
 🖹 Normal View🗖 Full Page View Calculator Precision (Decimal Places)0123456789101112131415 √xa The number in the first root (x) Power of the number in the first root (a) Index of the first root (m) √yb The number in the second root (y) Power of the number in the second root (b) Index of the second root (n)
Product of two roots Formula and Calculations Product of two roots, P = Quotient of two roots, Q = √xa ∙ √yb = √xan ∙ ybm√ ∙ √ = √ ∙ ∙ ∙ √ ∙ √ = √ ∙ √ ∙ √ = √ ∙ ∙ = √ = 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 √xa/√yb = √xan/ybm√/√ = √ ∙ / ∙ √/√ = √/√/√ = √// = √ = 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 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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## Theoretical description

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

xm = xm/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 = 26/3
= 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

xy = x ∙ y

and

x/y = x/y

This is true because

xy = x1/n ∙ y1/n = (x ∙ y)1/n = x ∙ y

and

x/y = x1/n/y1/n = (x/y)1/n = x/y

For example,

827 = 8 ∙ 27 = 216 = 6 because 63 = 216

Indeed,

8 = 2 and 27 = 3

Thus,

827 = 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

xayb = xa ∙ yb

and

xa/yb = xa/yb

This is true because

xayb = xa/n ∙ yb/n = (xa ∙ yb )1/n = xa ∙ yb

and

xa/yb = xa/n/yb/n = xa/yb1/n = xa/yb

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

xayb = xan ∙ ybm

This is true because

xayb = xa/m ∙ yb/n = xan/mn ∙ ybm/mn = (xan ∙ ybm )1/mn = xan ∙ ybm

and

xa/yb = xan/ybm

This is true because

xa/yb = xa/m/yb/n = xan/mn/ybm/mn = (xan/ybm)1/mn = xan/ybm

For example,

56 ∙ √28 = 3 ∙ 256 ∙ 2 ∙ 28 ∙ 3
= 512 ∙ 224
= 244,140,625 ∙ 16,777,216
= 4,096,000,000,000,000
= 400

Indeed,

56 = 56/3 = 52 = 25 and √(28 ) = 28/2 = 24 = 16

Thus,

56 ∙ √(28 ) = 25 ∙ 16 = 400

Another example,

45/1003 = 10 ∙ 645 ∙ 6/1003 ∙ 10
= 430/10030
= (4/100)30
= 430/60/10030/60
= 41/2/1001/2
= 4/100
= 2/10

Indeed,

45 = 45/10 = 41/2 = √4 = 2 and 1003 = 1003/6 = 1001/2 = √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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