Fractional indices (Higher)

Your phone can shift a song up by one semitone. Do that twelve times and you land exactly one octave higher, which is double the pitch. So twelve of those shifts multiply the pitch by two. Multiply, not add. Which means one shift multiplies by some number, and that number, multiplied by itself twelve times, gives exactly two. Two divided by twelve is not it. The number is two to the power one twelfth. About one point zero five nine. It is the spacing of every fret on a guitar, and it is what a fraction in an index does.

First, the bottom of the fraction, and why it means what it means.

You already have the index laws. Multiply powers of the same base and you add the indices. That is the rule from the video on multiplying and dividing powers, and it forces everything in this one. Take eight to the power one third. Multiply three of them together. The law says add the indices. A third plus a third plus a third is one. So you get eight to the power one, which is just eight. Look at what that says. Three copies of eight to the power one third multiply to give eight. The number that does that is the cube root of eight. So eight to the power one third is the cube root of eight, which is two.

Nothing new was invented there. The old law forced it. The bottom of the fraction tells you which root: a one half is a square root, a one third is a cube root, a one quarter is a fourth root.

Here is the line to carry into the exam. Bottom roots. Top powers. Root first.

Run that reading on twenty seven to the power one third. The bottom is a three, so it is asking for the cube root of twenty seven. Three times three times three is twenty seven, so the answer is three. Same reading, bigger number. A thousand to the power one third is the cube root of a thousand, which is ten.

Your turn. What is sixteen to the power one quarter? Is it two, is it four, or is it sixty four?

Have a think. I'll wait.

The answer is two. The bottom of the fraction is a four, so it is asking for the fourth root of sixteen.

Two times two times two times two is sixteen. Four is the square root, so four would be the answer to sixteen to the power one half, a different question. And sixty four is sixteen times four, which is the one thing an index never means.

Now the top of the fraction.

The other law you already have is power of a power. Raise a power to another power and the indices multiply. That is from the video on power of a power, and zero and negative indices. So write two thirds as one third, times two. Eight to the power two thirds is eight to the power one third, all squared. The one third takes the cube root. The two does the squaring.

Work it. The cube root of eight is two. Two squared is four. So eight to the power two thirds is four.

This is the step examiners watch go wrong. Here is how one report describes it.

Many were not able to interpret the fractional index and instead, incorrectly divided by 3 and multiplied by 4.

Divided by the bottom, multiplied by the top. It is written as a fraction, so it looks like division and multiplication. It is not. The bottom is a root and the top is a power.

One thing about the order. You are allowed to do the power first. Eight squared is sixty four, and the cube root of sixty four is four. Same answer.

Same answer, much bigger numbers on the way. Root first keeps everything small, so build that into your habit. It is a choice made for comfort, not a rule you can break.

And this notation reaches further than your hand would. Seven hundred and twenty nine to the power one sixth is three, because three multiplied by itself six times is seven hundred and twenty nine.

Your turn again. Sixteen to the power three quarters. Root first, then power.

Work that one out in your head. I'll wait.

The answer is eight.

The bottom is a four, so take the fourth root of sixteen, which is two. The top is a three, so cube it. Two cubed is eight. Small numbers all the way, because you rooted first.

Now put a minus sign in front of the fraction.

A negative index means the reciprocal, one over the power. That is from the same video on zero and negative indices, and nothing about it changes when the index happens to be a fraction. So a negative fractional index is two instructions stacked on top of each other. Deal with the fraction. Then flip.

Sixteen to the power minus one quarter. Instruction one is the fraction. The fourth root of sixteen is two. Instruction two is the minus sign. Flip it. One over two. So sixteen to the power minus one quarter is a half.

Tick them off one at a time on paper. Root and power first, then flip. What goes wrong is almost never the rooting or the flipping. It is doing one of them and forgetting the other.

So the line grows by one. Bottom roots. Top powers. Root first. Minus flips.

Time for the exam side of this.

An examiner report describes candidates meeting a negative fractional index and getting the two instructions tangled together.

A significant number, however applied the one over rule of dealing with a negative index to the fractional index itself, starting with thirty two to the power five thirds, from which no marks could be gained.

They pushed the minus sign into the fraction instead of treating it as its own step. Keep the two instructions separate on the page and that cannot happen to you.

A quick self check. If the base is bigger than one, a negative index always gives you an answer smaller than one. If yours came out big, you have missed the flip.

And watch the sign. A minus in the index never makes the answer negative. It flips it. Sixteen to the power minus one quarter is a half. It is never minus two.

Last one, and it uses both halves. Nine to the power minus one half.

Have a go at this one. I'll wait.

The answer is one third.

The bottom is a two, so square root of nine, which is three. The top is a one, so there is nothing to raise. Then the minus flips it. One over three, a third.

Time to fold all of that into one reading.

A fractional index is not new arithmetic. It is a root and a power, written in index notation, and both halves fall straight out of the index laws you already use. The bottom of the fraction is the root. The top is the power. Do the root first, because it keeps the numbers small. A minus sign in front flips the whole thing over.

One to try on paper. Twenty five to the power three halves.

Pause it there and work it out. I'll wait.

Root first, so the square root of twenty five is five, then cube it. A hundred and twenty five.

Bottom roots. Top powers. Root first. Minus flips.

Next in the chapter: converting to and from standard form, where powers of ten do the heavy lifting.

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