Power Play

Suppose a class message chain starts with one student who forwards a notice to 2 friends, each of whom forwards it to 2 more, and so on. In the tenth round the number of new messages is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Writing that out is painful and reading it back is worse. A power is the shorthand that was invented so that nobody has to.

We write that product as 2¹⁰, which equals 1024. In 2¹⁰, the number 2 is the base — the number being multiplied — and 10 is the exponent (also called the index or the power), which counts how many times the base appears as a factor. You read it as 2 raised to the power 10. The single most common mistake in this whole chapter is reading 2¹⁰ as 2 × 10 = 20. It is not 20. It is 1024, more than fifty times larger. Exponents are about repeated multiplying, never repeated adding.

A few values are worth knowing by heart. 3⁴ = 3 × 3 × 3 × 3 = 81. 5³ = 5 × 5 × 5 = 125. And 2¹⁰ = 1024, which is why one kilobyte was originally taken as 1024 bytes rather than 1000. Powers of 10 are the friendliest of the lot, because 10ⁿ is simply 1 followed by n zeros: 10³ = 1000, 10⁵ = 1,00,000 which is one lakh, and 10⁷ = 1,00,00,000 which is one crore. Keeping those two Indian landmarks in your head will make the later part of this chapter much faster.

Signs. When the base is negative, the exponent decides the sign of the answer. (-2)³ = (-2) × (-2) × (-2) = -8, while (-2)⁴ = 16. An even exponent pairs up the minus signs so they cancel; an odd exponent always leaves one minus behind. Watch the brackets very carefully here, because they change the meaning completely: (-3)² = 9, but -3² = -(3 × 3) = -9, since without brackets only the 3 is being squared and the minus sign is left outside.

Fractions as bases. The base does not have to be a whole number. (2/3)³ means (2/3) × (2/3) × (2/3) = 8/27, because the numerator and the denominator each get raised to the power. In the same way (3/4)² = 9/16.

Where you have already used powers. Prime factorisation is far neater in power form: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3², and 360 = 2³ × 3² × 5. Place value is really a statement about powers of ten. The number 47,258 can be written as 4 × 10⁴ + 7 × 10³ + 2 × 10² + 5 × 10¹ + 8 × 10⁰. Add those pieces up and you get 40000 + 7000 + 200 + 50 + 8 = 47,258, exactly as it should be. Every digit in every number you have ever written is riding on a power of ten, and that is precisely the idea that standard form pushes further later in this chapter.

Once you accept that a power is just repeated multiplication, the laws of exponents stop being rules to memorise. They become things you can rebuild on the spot, on the margin of your page, by writing the multiplication out in full.

1. Multiplying powers with the same base. Take 2³ × 2⁴. Written out it is (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s standing in a row, so it equals 2⁷ = 128. Check it the long way if you like: 8 × 16 = 128. The general rule is a^m × aⁿ = a^m+n. Same base multiplied, so add the exponents.

2. Dividing powers with the same base. Take 5⁶ ÷ 5⁴. The numerator carries six 5s and the denominator carries four 5s, so four of them cancel and two are left standing: 5² = 25. Checking the long way, 5⁶ = 15625 and 5⁴ = 625, and 15625 ÷ 625 = 25. The rule is a^m ÷ aⁿ = a^m-n, as long as a is not 0, since you can never divide by 0. Same base divided, so subtract the exponents.

3. A power raised to another power. (3²)³ means 3² × 3² × 3², which by the first law is 3²⁺²⁺² = 3⁶ = 729. Check it: 3² = 9, and 9³ = 9 × 9 × 9 = 729. The rule is (a^m)ⁿ = a^m×n. This one is worth extra care, because the natural instinct is to add the exponents when you should be multiplying them.

4. Same exponent, different bases. 2³ × 5³ = (2 × 5)³ = 10³ = 1000, which matches the direct calculation 8 × 125 = 1000. Division works in the same way: 6⁴ ÷ 3⁴ = (6 ÷ 3)⁴ = 2⁴ = 16, and the long check is 1296 ÷ 81 = 16.

Worked example. Simplify (2⁵ × 2²) ÷ 2³. Combine the top first using the product law: 2⁵⁺² = 2⁷. Then apply the quotient law: 2^7-3 = 2⁴ = 16.

Worked example with letters. Simplify (a⁵ × a³) ÷ (a²)³. The top gives a⁵⁺³ = a⁸. The bottom gives a^2×3 = a⁶. Dividing, a^8-6 = a². Letters behave exactly like numbers here, which is what makes these laws so useful in algebra later.

Two traps to avoid. First, the two laws that add or subtract exponents only work when the bases match, while the same-exponent law only works when the exponents match. The expression 2³ × 3² has neither, so it cannot be squeezed into a single power at all; you simply work it out as 8 × 9 = 72. Second, there is no law for adding powers. The value of 2³ + 2⁴ is 8 + 16 = 24, and 24 is nowhere near 2⁷ = 128. Addition needs real values, not exponent shortcuts. Whenever an expression has a plus or a minus sign joining two powers, stop and evaluate them separately.

So far every exponent has been a counting number. What could 5⁰ possibly mean — 5 used as a factor zero times? And what about 5^-2? The honest way to answer is to look at a pattern that already works and then insist that it keeps working.

The staircase. Write the powers of 10 going downwards: 10³ = 1000, 10² = 100, 10¹ = 10. Notice that every time the exponent drops by 1, the value is divided by 10. Keep walking down the same staircase. After 10 comes 1, so 10⁰ = 1. After 1 comes 1/10, so 10^-1 = 0.1. Then 10^-2 = 1/100 = 0.01 and 10^-3 = 1/1000 = 0.001. Nothing new was assumed anywhere; the same single step was just taken a few more times.

Why a⁰ = 1. There is a second argument that settles the matter completely. By the quotient law, 5³ ÷ 5³ = 5^3-3 = 5⁰. But 5³ ÷ 5³ is also just 125 ÷ 125 = 1. Both routes are correct, so 5⁰ must equal 1. The same argument works for any base except 0, which is why the rule is written as a⁰ = 1 for a ≠ 0. Notice that the size of the base makes no difference at all: 100⁰ = 1 and 7⁰ = 1 in exactly the same way. So 4⁰ + 5⁰ = 1 + 1 = 2. A very common slip is to add the bases first and write 9⁰ = 1, which is a different calculation and a wrong answer.

What a negative exponent means. A negative exponent does not make the number negative. It is an instruction to take the reciprocal. The rule is a^-n = 1/aⁿ. So 2^-3 = 1/2³ = 1/8, and 5^-2 = 1/5² = 1/25, and 10^-4 = 1/10000 = 0.0001. Every one of those is a small positive number. If you ever write 2^-3 = -8, that is the mistake this paragraph exists to prevent.

Negative exponents on fractions. Because a negative exponent flips things over, (2/3)^-2 = (3/2)² = 9/4. Turn the fraction upside down and the exponent becomes positive. Similarly (1/2)^-5 = 2⁵ = 32, and (2/5)^-3 = (5/2)³ = 125/8. This flip is the fastest route through most questions in this section.

Signs still follow the old rule. (-5)^-2 = 1/(-5)² = 1/25, which is positive because the exponent 2 is even. But (-2)^-3 = 1/(-2)³ = 1/(-8) = -1/8, which is negative because 3 is odd. Deal with the reciprocal and the sign as two separate decisions and you will not get tangled.

Worked example. Find (2^-1 + 3^-1)^-1. Work inside the bracket first: 2^-1 = 1/2 and 3^-1 = 1/3, and 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Now apply the outer exponent of -1, which simply flips the fraction: (5/6)^-1 = 6/5.

Worked example. Find x if 3^x = 1/27. Since 27 = 3³, we can write 1/27 = 3^-3, so x = -3. Best of all, every law from the previous section still applies word for word once negative exponents are allowed: 3^-2 × 3⁵ = 3^-2+5 = 3³ = 27, and 2⁵ × 2^-3 ÷ 2⁴ = 2^5-3-4 = 2^-2 = 1/4.

India's population is roughly 1,40,00,00,000 people. A state budget might be announced as ₹2,80,000 crore. The Sun sits about 14,96,00,000 km away. Numbers like these are hard to read, easy to mistype, and almost impossible to compare quickly. Standard form, also called scientific notation, fixes all three problems at once.

The shape of standard form. A number is in standard form when it is written as k × 10ⁿ, where k is at least 1 and less than 10, and n is an integer, so it may be positive, negative or zero. The condition 1 ≤ k < 10 is the part that makes the form standard: it forces exactly one non-zero digit to sit before the decimal point, so that every number has one and only one correct version. Without that condition, the same quantity could be written a dozen different ways and comparison would be no easier than before.

How to convert. Put the decimal point immediately after the first non-zero digit, then count how many places it has moved. Moving the point to the left means you have divided by that many tens, so you must multiply back by the same power of ten, and the exponent is positive.

Take 84,300. Written with its decimal point it is 84300.0. Move the point so that it sits just after the 8, giving 8.4300. The point travelled 4 places to the left, so 84,300 = 8.43 × 10⁴. Always check by going back the other way: 8.43 × 10000 = 84,300, which is what we started with.

Take 6,50,00,000, that is six crore fifty lakh. Placing the point after the 6 gives 6.5, and the point has moved 7 places, so the answer is 6.5 × 10⁷.

Real quantities. The distance from the Earth to the Sun, 14,96,00,000 km, becomes 1.496 × 10⁸ km. The speed of light, 30,00,00,000 m/s, becomes 3 × 10⁸ m/s. A population of 1,40,00,00,000 becomes 1.4 × 10⁹. The mass of the Earth is usually quoted as about 5.97 × 10²⁴ kg — try writing that one out in full and you will need 25 digits before you even reach the decimal point.

Lakh and crore as powers of ten. One lakh is 10⁵ and one crore is 10⁷, so any amount quoted in crores converts instantly. A sports contract of ₹16 crore is 16 × 10⁷, but 16 is not less than 10, so the correct standard form is ₹1.6 × 10⁸. A budget of ₹2,80,000 crore is 2.8 × 10⁵ × 10⁷ = ₹2.8 × 10¹², since 5 + 7 = 12.

The commonest error. Writing 34.5 × 10⁶ and calling it standard form. The value is perfectly correct, but the front number must sit between 1 and 10. Shift the point one more place to the left and add 1 to the exponent: 3.45 × 10⁷. The same tidying works the other way. If you end up with 0.45 × 10⁸, the front number is too small, so shift the point one place right and subtract 1 from the exponent to get 4.5 × 10⁷. Both of those equal 4,50,00,000, so nothing has been lost, only tidied.

Standard form is just as useful at the other end of the scale. The thickness of a sheet of paper, the width of a hair, the size of a bacterium — these are numbers with a long run of zeros sitting after the decimal point, and counting those zeros by eye is exactly how mistakes creep in.

The rule. Move the decimal point to the right until it sits just after the first non-zero digit, count the places you moved, and use that count as a negative exponent. Moving the point right has made the number bigger, so you must multiply by a matching small power of ten to bring it back to its true size.

Take 0.0016. Moving the point so that it sits after the 1 takes 3 places, so 0.0016 = 1.6 × 10^-3. Check by going back: 10^-3 = 0.001, and 1.6 × 0.001 = 0.0016.

Take 0.00045. Moving the point to sit after the 4 takes 4 places, so 0.00045 = 4.5 × 10^-4.

Take 0.000000000753. Moving the point to sit after the 7 takes 10 places, so the standard form is 7.53 × 10^-10. Nobody can count those zeros reliably by staring at them, which is precisely why this notation exists.

A quick way to check your work. In k × 10^-n, the first significant digit always lands in the n-th decimal place. So in 7 × 10^-5 the 7 must be the fifth digit after the point: 0.00007. Count them — 0, 0, 0, 0, 7 — and the 7 is indeed in fifth place. This single check catches almost every off-by-one slip, and it takes about three seconds.

Going back to usual form. To expand 3.02 × 10^-6, move the point 6 places to the left, filling the gaps with zeros: 0.00000302. Verify with the check above — the first significant digit, 3, should be in the sixth decimal place, and it is. To expand 5 × 10^-7, move 7 places to get 0.0000005.

Real quantities. A red blood cell is about 0.000007 m across, which is 7 × 10^-6 m. A human hair is roughly 0.00007 m thick, or 7 × 10^-5 m — exactly ten times thicker than the blood cell, and you can see that at a glance from the exponents alone without dividing anything. A typical bacterium measures about 5 × 10^-7 m. In science you will later meet the charge on an electron, about 1.6 × 10^-19 coulomb, and by then writing it out in full would be unthinkable.

Careful with the sign. A negative exponent tells you the number is small, not that it is negative. 2.5 × 10^-4 is 0.00025, a perfectly ordinary positive number that happens to be less than 1. And the more negative the exponent, the smaller the number: 10^-9 is a thousand million times tinier than 10⁰, and far tinier than 10^-3. Reading the exponent is the fastest way to judge size, which is the whole point of the notation.

Numbers in standard form are not just tidier to write. They are far easier to calculate with, because the front numbers and the powers of ten can be handled completely separately.

Multiplying. Multiply the front numbers, add the exponents, then tidy up if needed. (3 × 10⁵) × (2 × 10⁴) = (3 × 2) × 10⁵⁺⁴ = 6 × 10⁹. Sometimes the front number spills over 10, and then the tidying step matters: (4 × 10⁶) × (5 × 10³) = 20 × 10⁹, and since 20 = 2 × 10¹, the proper answer is 2 × 10¹⁰.

Dividing. Divide the front numbers and subtract the exponents. (8 × 10⁷) ÷ (2 × 10³) = 4 × 10⁴. Negative exponents behave in exactly the same way: (9 × 10^-4) ÷ (3 × 10²) = 3 × 10^-4-2 = 3 × 10^-6.

Adding and subtracting. Here you cannot work on the two parts separately, and skipping that fact is the single biggest source of wrong answers in this topic. First make both powers of ten the same, then add the front numbers. To find 3.2 × 10⁵ + 4.5 × 10⁴, rewrite the second term with the larger power: 4.5 × 10⁴ = 0.45 × 10⁵. Now 3.2 + 0.45 = 3.65, so the sum is 3.65 × 10⁵. Check the long way: 3,20,000 + 45,000 = 3,65,000, which is the same thing.

Comparing. Look at the exponent first, and only look at the front number if the exponents match. Between 3 × 10⁸ and 9 × 10⁷, rewrite the second one as 0.9 × 10⁸; now the contest is 3 against 0.9, so 3 × 10⁸ is the larger. Small numbers work identically. Which is bigger, 5 × 10^-6 or 4 × 10^-5? Rewrite 4 × 10^-5 as 40 × 10^-6, so 4 × 10^-5 wins comfortably, even though its front number is smaller.

Problem 1 — money per person. A state announces a budget of ₹2,80,000 crore for a population of 1.4 × 10⁸ people, that is 14 crore. First convert the budget: 2,80,000 crore = 2.8 × 10⁵ × 10⁷ = ₹2.8 × 10¹². Now divide: 2.8 ÷ 1.4 = 2, and 10¹² ÷ 10⁸ = 10⁴. The answer is 2 × 10⁴ = ₹20,000 per person, a figure you can actually picture.

Problem 2 — a stack of paper. One sheet is 1.2 × 10^-2 cm thick. How tall is a ream of 500 sheets? Write 500 as 5 × 10². Then (5 × 10²) × (1.2 × 10^-2) = (5 × 1.2) × 10^2-2 = 6 × 10⁰ = 6 cm. Notice how neatly the positive and negative exponents cancelled.

Problem 3 — storage. A 4 GB memory card holds about 4 × 10⁹ bytes. If one photo takes 5 × 10⁵ bytes, how many photos will fit? Divide: 4 ÷ 5 = 0.8, and 10⁹ ÷ 10⁵ = 10⁴, giving 0.8 × 10⁴. That front number is below 1, so tidy it: 0.8 × 10⁴ = 8 × 10³ = 8000 photos. The tidy-up step is not decoration — it is what turns a raw calculation into a properly written answer.