Arithmetic with roots

Principal value

When you manipulate roots and encounter difficulties with their multi-valuedness, it is best, to remove the multiple values at the start.

If a is a positive number, there is only one and only one positive value of x for which xⁿ = a. For x² = 4, the positive number is x = +2, for x⁴ = +81, it is x=+3, etc. In other words,

If a is positive, then has always one and only one positive value.

It is called the principal value. In what follows, we will assume that the numbers under the root sign are positive and understand always by their principal values.

will now not be +2 and -2, but only the principal value +2. If you require the negative value, you write - = -2, whence follows the Rule:

The equations = + x and (+x)ⁿ = a are equivalent - they have the same meaning.

If you replace a under the root sign by (+x)ⁿ, then . What is valid for positive terms and positive principal values under root signs also applies to absolute terms and absolute principal values.

Task: Find the error? Apparently, , whence -2 = +2.

The operations of taking a root and power of the same number do not cancel each other without further consideration, because .

Many amusing tasks with the root calculus are based on this or similar mistakes with leading signs. Similarly, it does now follow from (+2)² = (-2)² that +2 = -2; of course, it always follows from (a)ⁿ = (b)ⁿ that |a| = |b|!

Addition and subtraction of roots

Just like 3a + 4a, you can add . Only equal roots can be combined by addition and subtraction, that is, roots with the same number and same root exponent.

Examples:

1.
2.
3.

There may arise simplifications during multiplication and division when the roots are not equal and only their exponents agree.

Multiplication and division of roots with equal exponents

:

Take the nth power of the product, apply the power theorem (ab)ⁿ = aⁿ·bⁿ and obtain

.

The two equations (+x)ⁿ = a and +x = are equivalent, whence you can replace by .

I. Multiplication of roots with equal exponents:

Inversion:

Examples:

1.	2.
3.	4.
5.	6.

If a, b and c are negative, you find in 6. 2·|a|·|b|·|c|. We will accept this restriction in what follows without further reference.

7. Also the inverse transformation can be useful, if , for example, you can find in a table.

:

Take the nth power and apply the power theorem: (a/b)ⁿ = aⁿ/bⁿ:

Division of roots with equal exponents:

Inversion:

Examples:

1.				2.
3.				4.

Task: Compute 1/!

The computation can be simplified by making first the denominator rational by multiplication by a suitable number. You use and find

Examples:

1.

2.

3: You can make the denominator rational, using (a+b)(a-b)=a²-b².

.

Expand the fraction with :

4:

Powers of roots

Definition:

.

Application of Rule I yields

Rule III: The sequence of raising to a power and taking a root is arbitrary.

Examples:

1.
2.

The exponent of a power can also be negative:

Expansion and reduction of exponents, multiplication of roots with different exponents

Raise the equation to the power p and apply the Rule (a^m)ⁿ=a^mn, whence . Depending on whether you read the equation from the left to the right or in the opposite direction, you obtain:

Rule IV: .

Root and power exponents can be multiplied or divided by the same number without changing the value of the root.

Examples:

1.			2.
3.

Conversely:

4.

If you have a product or a fraction of roots with different exponents, this Rule lets you combine them by expansion or reduction of the exponents and application of Rules I and II:

Examples:

1.

2...

Roots of roots

Take first the mth and then the nth power :

whence

You would have arrived at the same result, if you had started from

Rule V. Multiple roots:

Examples:

1. You can manipulate or can compute or, since the sequence of taking roots is arbitrary,

2.

General tasks of taking roots

Summary of the five root rules

I.		II.		III.
IV.		V.

As with tasks involving powers, you must while taking roots be clear about every step in your calculation. Beside the definition, you can only use the above Rules.

Examples:

1. There is no rule for taking the root of a sum or a difference. You must compute 2³ and 3³ and then add them before taking the root:

2.

You could also have proceded as follows:

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