Index Laws Lab

An index just counts the multiplies

Indices look like a pile of separate rules to memorise. Add the powers here, subtract them there, a zero gives 1, a minus flips it over, a fraction turns into a root. Learned as a list it is easy to mix up under pressure. They are all one idea. An index counts how many times a base is multiplied by itself. In 2^5 the base is 2 and the index 5, and it means 2 multiplied by itself five times. Once you read an index as a count, every law is just bookkeeping on that count. Multiplying two powers of the same base joins the multiplications together, so you ADD the counts. Dividing cancels some off, so you SUBTRACT. A power of a power repeats the whole thing, so you MULTIPLY the counts. A zero index is no multiplications at all, which leaves 1. A negative index counts the other way, into dividing. And a fractional index counts a root. Standard form is the same counting made useful: it writes any number as a value from 1 to 10 times a power of ten, and the power of ten just counts the places. Carry one question through all of it: what is the index counting, and what does this operation do to the count?

What a power is made of

Five terms, each defined by what it is. The first two name the parts of a power; the rest are the special cases most often muddled.

Order how to crack a fractional index

Put the steps to work out 8 to the power two thirds into a sensible order.

• Read the fraction: the bottom, 3, is a root, and the top, 2, is a power
• Find the cube root of 8, which is 2
• Raise that result to the power 2
• Read off the value, which is 4
• Check it is sensible, since a root makes the number smaller before the power builds it back up

Which law finishes the sum

A student needs to simplify 2^6 divided by 2^2 and is not sure what happens to the indices. Which is right, and why?

• Subtract the indices, giving 2^4, because dividing cancels off some of the multiplications
• Divide the indices, giving 2^3, because the powers are being divided
• Add the indices, giving 2^8, the same as when multiplying
• Keep the index the same, giving 2^6, since the base does not change

Multiplying against dividing powers

Both laws work on the count of multiplications. The only difference is which way the count moves.

Work out a fractional index

Work out 27 to the power two thirds. The bottom of the fraction is the root, so find the cube root of 27 first, then raise it to the power 2. Give your answer as an ordinary number.

Three true rules of indices

Select the THREE statements about indices that are true.

• To multiply powers of the same base you add the indices
• Any base to the power zero equals 1
• A negative index means one over the base to the matching positive power
• To multiply powers of the same base you multiply the indices
• A power of a power is worked out by adding the two indices

What every index question is really asking

A shape that works on almost any indices question. Read the index as a count. It counts how many times the base is multiplied by itself. Say that to yourself before anything else. Then ask what the operation does to the count. Multiplying joins counts, so add them. Dividing cancels some, so subtract. A power of a power repeats, so multiply. These three cover most of the marks. Handle the special cases as counts too. A zero index is no multiplications, which leaves 1. A negative index counts into dividing, so it flips the power underneath 1. A fractional index counts a root on the bottom and a power on the top. Two habits cost marks. The first is changing the base: the base never changes under these laws, only the count does. The second is muddling the two multiplying rules: you ADD indices to multiply powers, but you MULTIPLY indices for a power of a power.

Match each power to what it equals

Numbers too big to write out

Here is an invented example, written for practice. A scientist notes down a very large number: 4 followed by seven zeros, which is 40000000. Numbers this size are clumsy to write and easy to miscount, so there is a tidier way. You keep the leading digit, 4, and then count how many places the point would move to turn 4 into 40000000. It moves seven places, so the number is 4 times ten multiplied by itself seven times, written 4 times ten to the power 7. It works for tiny numbers too. Take 0.0004. This time you count how many places the point moves the other way to turn 0.0004 into 4, which is three places, so the number is 4 times ten to the power minus 3. The sign of the power just says which way the point moved. The whole trick is counting places. A large number needs a positive power, a small number a negative one, and the digit at the front is always kept between 1 and just under 10.

The facts about powers

In a power such as 5^3, the number being multiplied, 5, is the _____, and the small number 3 is the _____. To multiply two powers of the same base you _____ the indices; to divide them you _____ the indices. Writing a number as a value from 1 to 10 times a power of ten is called _____ form.

Write a big number compactly

A number is written in standard form as 3.2 times ten to the power 4. Ten to the power 4 is 10000. Write the number in ordinary form.

Assemble the negative-index rule

Put together the rule for a negative index, using 4^-2 as the example.

Indices quick-fire

Five quick decisions on indices and standard form. Three lives.

Where the marks were lost

Three student answers on indices and standard form. The arithmetic is not the problem; the reasoning is.

• A student writes 3^2 times 3^4 as 9^6. What went wrong?
• A student writes 5^-2 as minus 25. What is the correct value?
• A student writes 47000 as 47 times ten to the power 3. Why is that not standard form?

Explain the one idea behind indices

A friend can work out simple powers but keeps getting the index laws, negative and fractional indices, and standard form wrong. Explain the single idea that ties indices together and how each rule follows from it.

• Explain what an index counts, using a simple power as an example
• Explain what happens to the indices when you multiply and when you divide powers of the same base
• Explain what a negative index means, with an example
• Explain what a fractional index means, with an example
• Explain what standard form is and the rule about the number at the front