Real Numbers, Exponents, and Radicals

This is Unit One of College Algebra, Part Two: roots and rational exponents. Part One ended with the exponent rules and the reason they take the form they do — each definition exists so that a rule already working keeps working. This part carries that story to its end, and the end is a genuinely surprising place. Roots start out looking like a separate topic with its own notation and its own rules. By the last entry they turn out to be exponents — fractional ones — and everything you learned in Part One applies to them without modification. Two errors get their own entries here, and both come from the same instinct: the assumption that a radical distributes over whatever it meets. It distributes over multiplication and division. It does not distribute over addition, and that one assumption is worth more marks than almost anything else in the unit. By the end you will be able to do five things. One: say why the square root of a square is a distance rather than the number itself. Two: say which operations a radical splits over, and which it never does. Three: simplify a square root, and know when you are finished. Four: convert between a fractional exponent and a root, both directions. Five: state the unit's story in one sentence.

One definition, and it is placed here rather than earlier for a reason — the next topic cannot be stated without it. It measures how far a number sits from zero, which throws away any information about which side.

The definition.

Alg 040 Absolute Value

How far a number lies from zero, so the answer is never negative. Negative four and positive four both sit four units away, on opposite sides, and both return four.

The radical symbol makes a choice, and the whole topic follows from it. Two numbers square to give twenty-five, and the symbol points at exactly one of them — which forces something odd when the thing under the radical is itself a square.

The choice.

Alg 041 The Principal Square Root

Both five and negative five square to twenty-five, and the radical returns five. It is the nonnegative one, always — the answer is never plus-or-minus.

What that forces.

Alg 042 The Square Root of a Square

Square any real number and take the root, and you get its distance from zero rather than the number back. Start at negative four, square it to sixteen, take the root, and you land on four.

And the error.

Alg 043 The Dropped-Absolute-Value Error

Writing the answer as the variable itself. For any negative value that comes out with the wrong sign — at negative four the shortcut gives negative four while the actual value is four.

Three entries, and the middle one is the most valuable thing in this lecture. A radical comes apart over multiplication and over division. It does not come apart over addition, and the arithmetic showing why takes about five seconds.

Over multiplication.

Alg 045 The Product Rule for Simplifying Square Roots

The root of a product equals the product of the roots, which is what lets a perfect square be pulled outside. Seventy-two becomes thirty-six times two, and the six comes out.

And not over addition.

Alg 046 The Split-Sum Error

The root of nine plus sixteen is the root of twenty-five, which is five. Rooting each separately gives three plus four, which is seven. Different answers, and the same trap sits inside any sum of two squares.

Over division.

Alg 048 The Quotient Rule for Simplifying Square Roots

The root of a fraction equals the root of the top over the root of the bottom — twenty-five over four becoming five over two. The bottom has to stay positive.

The procedure, and the useful part is the stopping condition — because with no rule for when to stop, it is not obvious that an answer is finished rather than merely partly done.

Watch one get simplified.

Alg 050 Simplifying a Square Root

Find the largest perfect square dividing into the number underneath, split the root across it, and write that square's root outside. Three hundred is a hundred times three. And when nothing above one divides in as a perfect square, it is already done.

One entry on combining radicals, and it is easier than it looks because it is the same move as collecting like terms — the radical behaves exactly like a variable you cannot evaluate.

When they combine.

Alg 052 Like Radicals

Three of something plus five of the same something is eight of it, and radicals follow that exactly. Different things underneath do not combine at all — they stay written as they are.

One procedure, and the thing to notice is why it is allowed: multiplying by a fraction whose top and bottom are identical multiplies by one, and multiplying by one changes nothing.

Watch it run.

Alg 054 Rationalizing a Denominator

Multiply the fraction by the radical over itself — a fraction equal to one, so the value is untouched while the radical moves upstairs. Both floors get multiplied; changing only one of them changes the number.

Roots of any order, and then the entry where the unit turns. Fractional exponents are introduced as new notation, and they are not new — they are the same roots written so that every rule from Part One applies to them unchanged.

Roots of any order.

Alg 056 The Principal nth Root

The number that raised to that power gives you what you started with. With an odd order a negative input is fine — the cube root of negative twenty-seven is negative three. With an even one, the nonnegative answer is taken.

The turn.

Alg 057 Rational Exponents: the 1/n Power

An exponent of one over something means exactly that root — eight to the one-third is the cube root of eight, which is two. The fraction upstairs was a root the whole time.

And the general form.

Alg 058 Rational Exponents: the m/n Power

The bottom of the fraction is the root and the top is the power. Sixteen to the three-quarters takes the fourth root, giving two, then cubes it to eight. Rooting first keeps the numbers manageable.

The unit closes by telling itself as one story, starting from counting. Each time an operation demanded an answer the current numbers could not supply, the numbers were extended — and every extension was made so that what already worked kept working.

The capstone. One story, told from the start.

Alg 060 From Counting to Rational Exponents

Subtraction demands answers counting cannot give, so the numbers extend. Division demands more, and so does taking roots. Each new kind of number and each new notation arrives so the earlier rules survive — and no claim in this entry is new.

Five things you can now do, and Unit One is complete. You can say why the square root of a square is a distance rather than the number itself, because the radical returns the nonnegative answer and a negative input would come back with the wrong sign. You can say where a radical splits — across multiplication and division — and where it never does. Nine plus sixteen gives five one way and seven the other. You can simplify a square root by pulling out the largest perfect square, and you know it is finished when nothing above one divides in as a square. You can move between a fractional exponent and a root in both directions, remembering that the bottom of the fraction is the root and the top is the power. And you can state the unit's story: every new number and every new notation was added so that rules already working kept working.