We use roots to express the inverse operation of exponentiation.

For example, if we have x², the square root helps us find the original number x when we know x². In other words, the square root of a number x is the value y that, when multiplied x by itself, gives y.

Basic rules of roots:

• Square Root: The square root of a number x is written as √x and means a number that when squared equals x. Ex. 3² = 3*3 = 9, 65² = 65*65 = 4225 ( Using the rule for numbers ending in 5: multiply the first part by its successor and append 25.)
• n-th Root: The n-th root of a number x is written as ⁿ√x and means a number that, when raised to the power n, equals x.
• Product Rule: √(a × b) = √a × √b
• Quotient Rule: √(a / b) = √a / √b, where b ≠ 0
• Root of a power: ⁿ√(x^m) = x^n/m
• Nested roots: ⁿ√(ᵐ√x) = ᵐⁿ√x = x^(1/(m·n))

Examples:

• √81 = 9 because 9 × 9 = 81
• √144 = 12 because 12 × 12 = 144
• √144*25 =√144 * √25 = 12 * 5 =60
• √225/25 = √225 / √25 = 25 / 5 = 5
• √(2² ) = 2^2/2 = 2¹ = 2
• ²√(⁵√x) = ^2*5√x = x^1/10

KEEP IN MIND

The square root symbol √, when written without an index, implicitly represents the second root.

For higher roots, the index n must be specified as √^[n]{x} = x^(1/n) to distinguish them.

To solve a nested roots problem in mathematics, follow these steps:

To solve the problem √√√1000

• Start with the innermost square root: √1000
  Calculate √1000 ≈ 31.6228
• Next, take the square root of the result: √31.6228
  Calculate √31.6228 ≈ 5.6234
• Finally, take the square root once more: √5.6234
  Calculate √5.6234 ≈ 2.3714

To solve √√√1000 by calculating roots together, first recognize that:

√√√1000 = √^2*2*2 (1000)= √⁸ (1000) = 1000^1/8

Now calculate:

1000^(1/8) = (10³)^(1/8) = 10^(3/8) ≈ 10^0.375 ≈ 2.3714

So, √√√1000 ≈ 2.3714 in both ways.

So, when dealing with nested roots, you need to multiply their indexes to simplify the expression correctly.

This works because taking the (m)-th root and then the (n)-th root is the same as taking the ((n \times m))-th root once. So, multiplying the indexes combines the nested roots into one root with the product of the original indexes.