Introduction to Exponents: Free Response

A power records two separate things, and the instruction supplies both of them. The number being used as a factor is $6$, and the number of times it is used is five.

Written as a power, that is

$6^5.$

Written out with nothing abbreviated, it is five copies of $6$ multiplied together:

$6^5 = 6 \times 6 \times 6 \times 6 \times 6.$

The base is the $6$, which is the number being multiplied. The exponent is the $5$, and it counts the copies.

Notice what does not appear in the full product: the $5$ itself. It is nowhere among the factors, because it was never a factor. It is a count of how many factors there are, and keeping that straight is the single most useful thing about this notation.

Take one factor at a time and carry the running product forward, so that no step ever asks you to hold more than two numbers at once.

$6 \times 6 = 36$

$36 \times 6 = 216$

$216 \times 6 = 1296$

$1296 \times 6 = 7776$

Five copies of the base needed four multiplications, because the first step uses two copies at once and every later step brings in one more. So

$6^5 = 7776.$

The other calculation is finished in a single step:

$6 \times 5 = 30.$

Put the two side by side and they are $7776$ and $30$. Multiplying the base by the exponent does not evaluate the power at all; it works out a different and far smaller quantity, and the distance between $30$ and $7776$ is the size of that mistake after only five factors.

The two names are not decorations. Each one records a measurement.

$6^2$ is read "six squared". A square with a side of $6$ units is made of six rows of six unit squares, and counting them is exactly this multiplication:

$6^2 = 6 \times 6 = 36.$

So a second power measures the area of a square built on that side, and that is where the word comes from.

$6^3$ is read "six cubed". A cube with an edge of $6$ units is built from six layers, each layer holding the $36$ unit cubes of a six by six square:

$6^3 = 6 \times 6 \times 6 = 216.$

So a third power measures the volume of a cube built on that edge.

There is no everyday fourth or fifth shape to borrow a word from, so past the third power the reading is plain: $6^5$ is read "six to the fifth power", or simply "six to the fifth". The pattern of names stops, but the meaning of the notation does not change one bit.

Take each power twice over, once from the definition and once from the rule.

For $2^2$ the definition asks for two copies of $2$:

$2^2 = 2 \times 2 = 4.$

The rule asks for the base multiplied by the exponent, which here is also $2 \times 2$:

$2 \times 2 = 4.$

The two agree, and it is worth seeing exactly why. With a base of $2$ and an exponent of $2$, the two recipes are written with the same two numbers joined by the same sign, so they could hardly have come out differently.

For $3^6$ the definition asks for six copies of $3$, brought in one at a time:

$3 \times 3 = 9, \qquad 9 \times 3 = 27, \qquad 27 \times 3 = 81$

$81 \times 3 = 243, \qquad 243 \times 3 = 729.$

The rule finishes in one step:

$3 \times 6 = 18.$

Here the two values are $729$ and $18$, which are not a near miss: the power is more than forty times what the rule reports.

Multiplication began as a shortcut for repeated addition, and $3 \times 6$ is three copies of $6$ added together. Since the order of two factors does not change a product, it is equally six copies of $3$ added together, and that is the form worth keeping here:

$3 \times 6 = 3 + 3 + 3 + 3 + 3 + 3 = 18.$

An exponent is the next shortcut along the same line, and it abbreviates repeated multiplication instead. Written out in full, that is

$3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729.$

Compare the two lines. Both start from six copies of $3$, and every copy is present in both. The only difference is what joins the copies together: plus signs on the first line, times signs on the second.

So the student's rule is not mis-evaluating the power. It is evaluating a different quantity altogether. It returns the repeated addition of the copies when the notation asked for their repeated multiplication.

That also accounts for the size of the gap in part A. Adding one more copy of $3$ moves a total up by three, while multiplying by one more copy of $3$ makes a total three times as large. Six steps of those two habits finish a very long way apart, and every further step widens the gap faster than the one before it.

The two kinds of case do not carry the same weight, and that difference is the whole question.

A rule of this sort is a claim about every power. One case where it agrees is consistent with the claim, but it is equally consistent with the claim being false, so on its own it establishes only that one instance and not the rule. One case where it disagrees is decisive, because the claim said every power: part A produced a power where the rule gives $18$ and the true value is $729$, and that single case is enough to reject it.

Agreeing cases are not even rare. Every first power is one, since a first power is a single copy of the base while the rule multiplies the base by $1$:

$8^1 = 8, \qquad 8 \times 1 = 8.$

The same happens for $9^1$, for $57^1$, and for any base whatever. A base of $0$ agrees at every positive whole-number exponent as well, since multiplying copies of $0$ and multiplying $0$ by the exponent both give $0$. So a supply of agreements can be manufactured without the rule holding anywhere else.

The verdict is that the rule is false, and the case that first persuaded the student was one of those accidents: with a base of $2$ and an exponent of $2$, the two recipes are the same multiplication written twice. What the rule is worth is as a reminder of what an exponent is not. It counts the factors; it is not one of them.

Follow the chain one step at a time. Each step multiplies the running count of seeds by ten.

A sachet holds $10$ seeds. A packet holds ten sachets, so a packet holds

$10 \times 10 = 100$

seeds. A box holds ten packets, a case holds ten boxes, and a pallet holds ten cases, which brings in three more factors of ten:

$10 \times 10 \times 10 \times 10 \times 10 = 100{,}000.$

Five tens are multiplied together, and that is exactly what the notation $10^5$ records, so a pallet holds

$10^5 = 100{,}000$

seeds.

Now count the digits of that value. There are six: a leading $1$ and then five zeros. The five zeros are the five packing steps, one for every factor of ten, and the leading $1$ is the single seed the chain started from. Keep those two counts apart, because the exponent matches the zeros rather than the digits.

Every factor of ten in a power of ten appends one zero, because multiplying a whole number by ten shifts each of its digits one place up and writes a $0$ into the ones place. So the zeros can be counted instead of the multiplications being done.

The total is written

$100{,}000{,}000,$

which is a $1$ followed by eight zeros. Eight zeros mean eight factors of ten:

$10^8 = 100{,}000{,}000.$

No multiplication was carried out anywhere in that. The journey back is just as quick: to turn a power of ten into its value, write a $1$ and then as many zeros as the exponent names.

Write the clerk's own number out before judging anything. Nine factors of ten give a $1$ followed by nine zeros:

$10^9 = 1{,}000{,}000{,}000.$

Count its digits: the leading $1$ and then nine zeros make ten digits, not nine. So the clerk's example does not even have the property they claimed for it.

The relationship is off by one all the way along. For a whole-number exponent $n$ of at least $1$, the value of $10^n$ is a $1$ followed by $n$ zeros, so it carries $n$ zeros and $n + 1$ digits. The nine-digit power of ten is therefore the one with eight zeros:

$10^8 = 100{,}000{,}000,$

which is the annual total from part B.

There is a second thing wrong with the statement, and it is the larger of the two. Nine-digit counts run all the way from $100{,}000{,}000$ to $999{,}999{,}999$, and almost none of them are powers of ten: a count of $384{,}512{,}907$ seeds has nine digits and is no power of ten. Knowing the digit count of a number tells you roughly how big it is. It does not turn it into a power of ten.

Build the count of slabs first, and price it afterwards.

The courtyard is a square grid with twelve slabs along each side, so it holds twelve rows of twelve slabs, which is a second power:

$12^2 = 12 \times 12.$

The fountain covers a square patch five slabs along each side, which is five rows of five:

$5^2 = 5 \times 5.$

Those slabs are not repaved, so the number that is repaved is the first count with the second taken away, and the price of $8$ dollars applies to that whole difference:

$8 \times (12^2 - 5^2).$

The bracket is doing real work. Each power is settled before the subtraction in any case, since exponents outrank subtraction, but without the bracket the multiplication by $8$ would also come before the subtraction, and only the first count would be priced.

Taking the expression as it stands, nothing outside the bracket may act until the bracket has become a single number, so start inside it. (Rewriting the expression before evaluating it is a different route, and the alternate method takes it.) Inside, the powers sit on the higher tier and are settled first:

$12^2 - 5^2 = 144 - 5^2$

$144 - 5^2 = 144 - 25$

Now the subtraction, which is the only operation left inside the bracket:

$144 - 25 = 119.$

That $119$ is the number of slabs being repaved. The whole expression has become a single multiplication:

$8 \times 119 = 952.$

The repaving costs $952$ dollars.

Evaluate the line exactly as it is written. The bracket is settled first, then the power, then the multiplication:

$8 \times (12 - 5)^2 = 8 \times 7^2$

$8 \times 7^2 = 8 \times 49 = 392.$

So that line asks for $392$ dollars, against the $952$ dollars the job actually costs, leaving the contractor $560$ dollars short.

The $49$ is easy to picture, and picturing it is the quickest way to see what went wrong. It is a square patch seven slabs on a side, the square you would get by shortening both sides of the courtyard by the width of the fountain and throwing the rest away. The courtyard with the fountain patch removed is not that shape at all: it is a twelve by twelve grid with a five by five hole in the middle of it, and it holds $119$ slabs.

The difference is entirely about what the exponent is attached to. In $12^2 - 5^2$ each exponent is attached to a single side length, so two square counts are formed and one is subtracted from the other. In $(12 - 5)^2$ the exponent is attached to the difference of the two side lengths, and it squares a length that no side of the repaved region has. Parentheses decide what a power applies to, and moving them here moved the shape being counted.

The parentheses are the only thing in either expression that says what the exponent is attached to, so settle that before multiplying anything.

In $(-7)^2$ the parentheses bundle the sign into the base, so the base is the single number $-7$, and squaring it means multiplying it by itself:

$(-7)^2 = (-7) \times (-7) = 49.$

There are two negative factors, and a negative times a negative is positive, so the result is positive.

In $-7^2$ there are no parentheses, so by the order of operations the exponent binds to the $7$ alone, ahead of the minus sign standing in front. Read it as $-(7^2)$, which is to square first and negate afterwards:

$-7^2 = -(7 \times 7) = -49.$

Only one minus sign is in play here and it never enters the multiplication at all, so the result stays negative. Same digits, same exponent, opposite signs, and the parentheses are the whole of the difference.

Take the parenthesised one first. The base is $-7$ and the exponent calls for three copies of it:

$(-7)^3 = (-7) \times (-7) \times (-7).$

The first two factors multiply to $49$, and the third brings the sign back in:

$49 \times (-7) = -343.$

Now the one without parentheses. The exponent binds to the $7$ alone and the minus sign waits its turn:

$-7^3 = -(7 \times 7 \times 7) = -343.$

Both expressions come to $-343$, so at this exponent they agree. They got there by different routes: in the first, three negative factors leave one minus sign uncancelled, while in the second a positive cube is negated at the end. This time the two routes land on the same number.

The claim says always, so a single exponent where the two agree is enough to reject it, and part B supplied one: at the third power both expressions came to $-343$.

What is really going on is a count of minus signs. Take a positive base and a positive whole-number exponent.

In the parenthesised version the whole negative number is the base, so every factor in the product carries a minus sign. Negative factors cancel in pairs, so an even count of them leaves the product positive, while an odd count leaves exactly one minus sign uncancelled:

$(-7)^2 = 49, \qquad (-7)^3 = -343.$

In the version without parentheses there is exactly one minus sign however large the exponent, and it is applied after the power has been worked out, so the result is negative every time:

$-7^2 = -49, \qquad -7^3 = -343.$

Set the two lines beside each other. When the exponent is even, the first value is positive and the second is negative, so they differ. When the exponent is odd, the first value is negative and the same size as the second, so they agree. For a positive base and a positive whole-number exponent that is the whole story: the two arrangements agree exactly when the exponent is odd.

So the parentheses are neither decoration nor always decisive. They decide whether the sign sits inside the multiplication or outside it, and that choice only shows up in the answer when an even count of negative factors would have cancelled.