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Radicals

A radical is a rational exponent wearing a different hat, so nothing here is new mathematics — it is the previous section with a sign drawn over it. What is new is the convention: there is an agreed standard for what a finished radical answer looks like, and most of the work in this section is getting an expression into that form.

Class Notes — Chapter 1, Section 1.3

Calculator
Anatomy and conversion
The index sits in the notch of the radical sign, the radicand sits under the bar. A missing index always means 2.
  • Radical to exponent

    The index becomes the denominator. Converting is usually the fastest way to start a hard problem.

  • Radical of a power

    This is the bridge between the two sections. Every radical rule below is an exponent rule read through it.

  • Principal root

    For an even index the radical means the non-negative root only. , never — the belongs to solving , not to the symbol.

  • Odd index, negative radicand

    An odd index handles negatives happily. An even index of a negative radicand is not a real number.

The laws
Products and quotients split. Sums and differences do not — that is the whole cautionary tale.
  • Product

    Works in both directions: split a radicand to simplify it, or join two radicals of the same index to multiply.

  • Quotient

    Same index top and bottom. Indexes must match before anything can be combined.

  • Odd index undoes an odd power

    Nothing to worry about: the sign passes straight through.

  • Even index needs absolute value

    , not . This is the single rule the "assume positive" instruction exists to switch off.

  • Root of a root

    Because . Multiply the indexes.

  • No rule for sums

    There is no way to split a sum under a radical. while .

What counts as simplified
An answer is not finished because it is shorter. Three conditions have to hold, and the problems in this section are each one of them being enforced.

  1. No factor under the radical has an exponent as big as the index. is not simplified because . Pull out pairs until at most one is left underneath.

  2. No fraction under the radical. becomes . Split it with the quotient rule and resolve the bottom.

  3. No radical in the denominator. This is what rationalizing is for. It is a convention rather than a law — but it is the convention every answer key uses.

  4. The index is as small as it will go. . Convert to exponent form, reduce the fraction, convert back.

The extraction rule
Simplifying a radical is one division, done once per factor. Everything else is bookkeeping.

  1. Divide the exponent by the index. For , write . The quotient is how many 's come out; the remainder is how many stay under.

  2. Do it separately for every factor. In : gives , so one 2 out and one 2 in; , so out and in.

  3. Break numbers into prime powers first. You cannot see what comes out of until it is . Then and a single 2 steps out.

  4. Multiply everything together before extracting. When several radicals of the same index are multiplied, join them into one radicand first. Extracting from each separately misses the factors that only pair up once combined.

Worked examples — simplifying and multiplying
Every variable below is assumed positive, so no absolute value bars are needed.

Rationalizing: which move to use
Three different denominators, three different multipliers. Picking the right one first is most of the battle.

  1. One square root on the bottom. Multiply top and bottom by that same root. and it is done.

  2. One root of index on the bottom. You need each exponent under the radical to reach . For multiply by , because and . Multiplying by the same radical again is the classic mistake — it only works when the index is 2.

  3. Two terms on the bottom. Multiply by the conjugate — the same two terms with the middle sign flipped. squares both terms at once, which is what clears them.

  4. Leave the numerator factored. Expanding the top after conjugating usually makes the answer longer and no clearer. The denominator is the part that had to change.

Worked examples — rationalizing

True or false

  • The exponent is attached to the alone, so only the goes under the radical. Test at : , while .

  • Radicals distribute over multiplication, never over addition — the same wall as , seen from the other side. Test at : , while .

  • . Without the positivity assumption it would be , since the left side is defined for negative and the right side is not.

Why the variables are assumed positive
Every simplifying and rationalizing problem in this section carries the instruction "assume , and are positive". It is not decoration — here is exactly what it buys, and what the answers would become without it.

  1. It removes the absolute value bars. For an even index, . Without the assumption, would still be fine — is never negative — but would need -style care wherever an odd power came out from under an even index.

  2. It guarantees the radical exists. is not a real number when is negative. Without the assumption, every answer would have to be stated with a domain attached.

  3. It keeps rationalizing legal. Multiplying by assumes is a real non-zero number. If could be negative or zero, that step would not be multiplication by 1.

  4. What would change. Answers would carry absolute values on any factor extracted from an even-index radical with an odd exponent, and each result would be quoted on a restricted domain. The algebra itself would not change at all — only the bookkeeping around the sign.

Practice
Assume every variable is positive.

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