Power Play
Why This Matters
Take a sheet of paper. Fold it in half. Fold it again. And again.
How many times can you keep folding? Try it. Most people get stuck at about 7 folds, even with thin paper. But here is a wild question: imagine you could fold it as many times as you wished. After 46 folds, how thick would the paper be?
A few centimetres? A metre? A whole building’s height?
The real answer is shocking. After just 46 folds, the paper would be thick enough to reach the Moon — over 7,00,000 km. That is not a typo. Forty-six folds.
How can something so tiny grow so huge, so fast? The secret is that the thickness doubles every fold. Doubling, again and again, builds numbers our brains simply cannot picture. This is called exponential growth, and it is everywhere — in money that grows with interest, in a virus spreading, in computer storage, in the number of stars in the sky.
To handle such huge numbers (and also tiny ones, like the size of an atom), we need a smart shorthand. That shorthand is exponents and powers. By the end of this chapter, you will write a number as long as 5,97,60,00,00,00,00,00,00,00,00,00,000 kg in just a few symbols — and read it back without losing a single zero.
The Big Idea
An exponent is a shortcut for repeated multiplication, just like multiplication is a shortcut for repeated addition. Writing 2⁴ instead of 2 × 2 × 2 × 2 is not just shorter — it lets us handle numbers that are far too big or far too small to write out. Once we know this shorthand, a small set of simple rules (the laws of exponents) lets us multiply, divide and compare these giants and midgets with ease. And powers of 10 give us a ruler for measuring the whole universe, from a grain of sand to the number of stars.
Before we go further, let’s actually see why the folded paper explodes in size — the picture below tracks the thickness fold by fold.
Let’s Break It Down
We will go in order. First, what a power actually means. Then the laws of exponents — and why each one is true. Then the zero and negative exponents. Then powers of 10, and finally scientific notation, the language scientists use for huge and tiny numbers.
Before we start, let us quickly remember squares and cubes from earlier classes, because powers grow straight out of them.
What a power really means
We just saw a square uses a number twice and a cube uses it three times. Why stop there? We can multiply a number by itself any number of times.
- n × n × n × n = n⁴ (read “n to the power 4”)
- n × n × n × n × n × n × n = n⁷ (read “n to the power 7”)
In general, nᵃ means n multiplied by itself a times.
Every power has two parts. Let’s name them on a real example: 5⁴.
The picture below labels the parts clearly.
So in 5⁴ = 5 × 5 × 5 × 5 = 625, the base is 5, the exponent is 4, and 625 is the value. We call 5¹, 5², 5³, 5⁴, … the powers of 5.
A few more examples, including negative bases:
- 4 × 4 × 4 = 4³ = 64
- (−4) × (−4) × (−4) = (−4)³ = −64 (an odd number of negative signs gives a negative answer)
- a × a × a × b × b = a³b² (read “a cubed b squared”)
Let’s use powers to break a big number into its prime building blocks.
Express 32400 as a product of its prime factors, written in exponential form.
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Keep dividing 32400 by the smallest prime that fits. Divide by 2: 32400 = 2 × 16200 = 2 × 2 × 8100 = 2 × 2 × 2 × 4050 = 2 × 2 × 2 × 2 × 2025. Now 2025 is odd, so 2 is finished — we used four 2s.
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Now factor 2025. Divide by 5: 2025 = 5 × 405 = 5 × 5 × 81. So we used two 5s, and we are left with 81.
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Factor 81. Divide by 3: 81 = 3 × 27 = 3 × 3 × 9 = 3 × 3 × 3 × 3. So we used four 3s.
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Collect everything: 32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3. Count each prime and write it as a power: four 2s, two 5s, four 3s. So 32400 = 2⁴ × 5² × 3⁴. Quick check: 16 × 25 × 81 = 32400. Correct.
Is (−1)⁵ positive or negative? What about (−1)⁵⁶? And is (−2)⁴ equal to 16?
(−1)⁵ means −1 multiplied 5 times. An odd number of minus signs gives a negative answer, so (−1)⁵ = −1 (negative). (−1)⁵⁶ has an even number of minus signs, so the answer is +1 (positive). And (−2)⁴ = (−2)(−2)(−2)(−2) = 4 × 4 = 16 — yes, because four minus signs (even) cancel out to give a positive answer.
The product law: multiplying powers with the same base
Here is a story from the textbook. A king gives each of his 3 daughters 3 baskets. Each basket has 3 keys, each key opens 3 rooms, each room has 3 tables, each table has 3 necklaces, and each necklace has 3 diamonds. How many diamonds in all?
Count the multiplications: 3 (daughters) × 3 (baskets) × 3 (keys) × 3 (rooms) × 3 (tables) × 3 (necklaces) × 3 (diamonds) = 3⁷.
Now suppose you had already worked out the number of rooms, which is 3⁴ = 81. And you know 3³ = 27. Can you reach 3⁷ quickly? Yes:
3⁷ = (3 × 3 × 3 × 3) × (3 × 3 × 3) = 3⁴ × 3³
Look closely. The first bracket has 3 written 4 times. The second has 3 written 3 times. Put together, that is 3 written 4 + 3 = 7 times, which is 3⁷. The exponents simply added up.
This always works. For any base, just count the total number of factors:
aᵐ × aⁿ = aᵐ⁺ⁿ (same base: ADD the exponents)
It works for letter-numbers too. For example, p⁴ × p⁶ = (p×p×p×p) × (p×p×p×p×p×p) = p¹⁰, and 4 + 6 = 10. The rule is not magic — it is just counting how many times the base appears.
Simplify 2⁻⁴ × 2⁷ and write the answer as a single power of 2.
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The base is the same (2), and we are multiplying. So we use the product law: add the exponents. (This works even when one exponent is negative — more on negative exponents soon.)
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Add the exponents: (−4) + 7 = 3.
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So 2⁻⁴ × 2⁷ = 2³. (And 2³ = 8, if you want the value.)
The power-of-a-power law
Look at 4⁶. We can group its six 4s in different ways.
Group them in twos — three groups of (4 × 4):
4⁶ = (4 × 4) × (4 × 4) × (4 × 4) = 4² × 4² × 4² = (4²)³
Or group them in threes — two groups of (4 × 4 × 4):
4⁶ = (4 × 4 × 4) × (4 × 4 × 4) = 4³ × 4³ = (4³)²
In both cases we get 4⁶. Notice the exponents multiply: in (4²)³, we have 2 × 3 = 6; in (4³)², we have 3 × 2 = 6. A “power of a power” means you have that many groups of that many factors, so you multiply to get the total count.
(aᵐ)ᵇ = aᵐ×ᵇ (power of a power: MULTIPLY the exponents)
This also tells us a number can often be written as a power of a power in several ways. For example, 8⁶ = (8²)³ = (8³)², and since 8 = 2³, we can even write 8⁶ = (2³)⁶ = 2¹⁸.
Write 7¹⁵ as a power of a power in two different ways.
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We need two numbers that multiply to 15, because the power-of-a-power law multiplies exponents. The factor pairs of 15 are 3 × 5 and 5 × 3 (and 1 × 15).
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Using 3 × 5: split 15 as 3 × 5, so 7¹⁵ = (7³)⁵. Check: 3 × 5 = 15. Correct.
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Using 5 × 3: split 15 as 5 × 3, so 7¹⁵ = (7⁵)³. Check: 5 × 3 = 15. Correct. So two valid answers are 7¹⁵ = (7³)⁵ = (7⁵)³.
Same exponent, different bases
Here is another textbook puzzle. In one pond, lotuses double each day; in another, they triple. Start one lotus in the doubling pond. After 4 days it becomes 2⁴. Move those into the tripling pond for 4 more days, and each gets tripled 4 times: 2⁴ × 3⁴.
Now write it out and regroup:
2⁴ × 3⁴ = (2 × 2 × 2 × 2) × (3 × 3 × 3 × 3)
= (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) = (2 × 3)⁴ = 6⁴
We just paired each 2 with a 3. Since there are four 2s and four 3s, we get four pairs of (2 × 3). So:
aⁿ × bⁿ = (a × b)ⁿ (same exponent: multiply the bases)
The same idea works for division: aⁿ ÷ bⁿ = (a ÷ b)ⁿ. For example, 10⁴ ÷ 5⁴ = (10 ÷ 5)⁴ = 2⁴ = 16.
The quotient law: dividing powers with the same base
Imagine a line 16 units long, which is 2⁴ units. Erase half of it — you remove half the factors:
2⁴ ÷ 2 = (2 × 2 × 2 × 2) ÷ 2 = 2 × 2 × 2 = 2³ = 8
Halve it again: 2⁴ ÷ 2² = (2×2×2×2)/(2×2) = 2 × 2 = 2² = 4. And again: 2⁴ ÷ 2³ = (2×2×2×2)/(2×2×2) = 2 = 2¹.
See the pattern? Each time we divide, the matching factors cancel from top and bottom, and the exponents subtract:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base: SUBTRACT the exponents, with a ≠ 0)
So 2⁴ ÷ 2³ = 2⁴⁻³ = 2¹, exactly what we got. And a bigger one: 2¹⁰⁰ ÷ 2²⁵ = 2¹⁰⁰⁻²⁵ = 2⁷⁵.
Why a⁰ = 1
What if the exponents are equal, like 2⁴ ÷ 2⁴? The answer is obviously 1 (any non-zero number divided by itself is 1). But the quotient law says 2⁴ ÷ 2⁴ = 2⁴⁻⁴ = 2⁰. So to keep the law working, we must define:
a⁰ = 1, for any a ≠ 0
This is not a random rule. It is the only value that keeps the laws of exponents consistent. Let’s see it spelled out: 2⁰ = 2⁴⁻⁴ = 2⁴ ÷ 2⁴ = 16 ÷ 16 = 1. (Why must a not be 0? Because 0⁰ would mean 0 ÷ 0, which is not defined.)
Negative exponents — the other side of powers
Keep halving past 1. Take the 16-unit (2⁴) line and halve it 5 times:
2⁴ ÷ 2⁵ = (2×2×2×2)/(2×2×2×2×2) = 1/2
By the quotient law, 2⁴ ÷ 2⁵ = 2⁴⁻⁵ = 2⁻¹. So 2⁻¹ = 1/2.
Halve 2⁴ ten times: 2⁴ ÷ 2¹⁰ = 2⁴⁻¹⁰ = 2⁻⁶. Written out, that is 1/2⁶ = 1/64. So 2⁻⁶ = 1/2⁶.
The pattern is clear. A negative exponent means “flip it to the bottom” — take the reciprocal:
a⁻ⁿ = 1/aⁿ (negative exponent: reciprocal)
aⁿ = 1/a⁻ⁿ (and the same idea works in reverse)
So 10⁻³ = 1/10³ = 1/1000, and 7⁻² = 1/7² = 1/49. A negative exponent does not make the number negative — it makes it a small fraction.
With this, all the laws now work for any integer exponents, positive, zero or negative. Here is the whole toolkit in one place.
Simplify (13⁻²)⁻³ and write the answer as a power of 13.
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This is a power of a power, so we multiply the exponents — even when they are negative.
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Multiply: (−2) × (−3) = +6. (A negative times a negative is positive.)
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So (13⁻²)⁻³ = 13⁶.
Powers of 10 — a ruler for place value
You already write numbers in expanded form. Powers of 10 make that even neater. For example:
47561 = (4 × 10000) + (7 × 1000) + (5 × 100) + (6 × 10) + 1
= (4 × 10⁴) + (7 × 10³) + (5 × 10²) + (6 × 10¹) + (1 × 10⁰)
Each place is just the next power of 10. And it keeps working past the decimal point, using negative powers. The picture below shows this for 561.903.
So 10⁰ = 1 sits exactly at the ones place — another reason the rule a⁰ = 1 makes sense.
Scientific notation — the language of huge and tiny numbers
Some real numbers are monstrous. The mass of the Earth is about 59,76,00,00,00,00,00,00,00,00,00,000 kg. Count those zeros — go on, try. You will lose track. Miscounting by one zero turns ₹5,000 into ₹50,000. With these giants, the number of zeros matters more than the front digits.
So we use a smart shorthand. The trick: write any number as (a number between 1 and 10) × (a power of 10).
Watch how 5900 can be shifted:
5900 = 590 × 10¹ = 59 × 10² = 5.9 × 10³ = 0.59 × 10⁴
Each time we move the decimal one place left, the number shrinks by 10, so we multiply by one more 10 to keep it equal. The version where the front number sits between 1 and 10 is the special one. That is scientific notation (also called standard form):
A number written as x × 10ʸ, where x ≥ 1 and x < 10, and y is any integer.
Here x is the coefficient and y is the exponent. The exponent is the star: it tells you the size. When Mumbai’s population (about 2 crore) is written as 2 × 10⁷, the 7 matters most — change it to 8 and the population jumps from 2 crore to 20 crore. The picture below shows the parts and a number line of sizes.
So the Earth’s mass becomes a tidy 5.976 × 10²⁴ kg — same value, far easier to read and compare.
Let’s convert a number into standard form carefully.
Write 70,04,00,00,000 in scientific (standard) form.
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First read all the digits: 7 0 0 4 0 0 0 0 0 0 0. Strip the commas and write them in order: 70040000000.
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We want one non-zero digit before the decimal point. Place the decimal right after the first digit: 7.004. The remaining digits become the decimal part.
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Count how many places the decimal moved from the end of the whole number to its new spot. The original 70040000000 has 11 digits, so the decimal moves 10 places left. That gives the exponent 10.
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So 70,04,00,00,000 = 7.004 × 10¹⁰. (Check: 7.004 × 10¹⁰ shifts the decimal 10 places right, rebuilding 70040000000. Correct.)
The distance from the Sun to Saturn is 1.4335 × 10¹² m, and from the Sun to Earth is 1.496 × 10¹¹ m. Which distance is smaller?
Compare the exponents first, because they tell you the size. Saturn’s distance has exponent 12; Earth’s has exponent 11. A smaller exponent means a smaller number (10¹¹ is ten times smaller than 10¹²). So the Sun–Earth distance, 1.496 × 10¹¹ m, is smaller — even though its coefficient 1.496 is bigger than 1.4335. With scientific notation, always check the exponent before the coefficient.
Multiplying numbers already in scientific form
To multiply numbers in scientific form, multiply the coefficients together and add the exponents (that is the product law for the powers of 10).
There are about 100 million bee colonies in the world, each with about 50,000 bees. Find the total number of bees in scientific notation.
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Write each number in scientific form. 100 million = 10⁸. And 50,000 = 5 × 10⁴.
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Multiply: total bees = 10⁸ × 5 × 10⁴ = 5 × (10⁸ × 10⁴).
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Add the exponents of 10: 8 + 4 = 12. So 10⁸ × 10⁴ = 10¹².
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So the total is 5 × 10¹² bees — five trillion bees. (In ordinary digits that is 5 followed by 12 zeros; scientific notation says it in three symbols.)
Common Mistakes
These three slip-ups trip up almost everyone at first. Spot them now and you will avoid easy marks lost in exams.
To work out 2⁵, you multiply the base by the exponent: 2 × 5 = 10.
Exponents are written right next to the base, just like a multiplication, so the eye reads '2 and 5' and wants to multiply them — the same move that works for 2 × 5.
The exponent counts repeated multiplication of the base by itself, not the base times the exponent. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, not 10.
A negative exponent makes the number negative, so 2⁻³ must be a negative number like −8.
The minus sign sits right there in front of the 3, and minus signs usually do mean 'negative', so it feels like the whole answer should turn negative.
A negative exponent means take the reciprocal — flip the number to the bottom. 2⁻³ = 1/2³ = 1/8, which is a small positive fraction. The sign of the answer never comes from the exponent.
When you multiply powers with the same base, you multiply the exponents: 3² × 3⁴ = 3⁸.
There is a real rule where exponents multiply — the power-of-a-power rule (3²)⁴ — and it is easy to mix it up with plain multiplication of two separate powers.
For a product of powers with the same base, you ADD the exponents: 3² × 3⁴ = 3²⁺⁴ = 3⁶. Exponents multiply only for a power of a power, like (3²)⁴ = 3⁸.
Quick Check
Try each question. The explanation appears after you answer, so read it either way.
What is the value of 3⁴?
3⁴ means 3 multiplied by itself 4 times: 3 × 3 × 3 × 3 = 9 × 9 = 81. (12 is the mistake of doing 3 × 4; 27 is only 3³; 64 is 4³.)
Simplify 5⁷ ÷ 5⁴ as a single power of 5.
Same base, dividing — so subtract the exponents: 5⁷ ÷ 5⁴ = 5⁷⁻⁴ = 5³. (5¹¹ would be adding instead of subtracting.)
What is the value of 10⁻²?
A negative exponent means the reciprocal: 10⁻² = 1/10² = 1/100 (which is 0.01). It is a small positive fraction, never a negative number.
Which of these is 65,950 written correctly in standard form?
In standard form the coefficient must be between 1 and 10. Move the decimal so it sits after the first digit: 6.595. The decimal moved 4 places, so the exponent is 4: 6.595 × 10⁴. (65.95 × 10³ has a coefficient bigger than 10, and 0.6595 × 10⁵ has one smaller than 1 — both break the rule.)
Practice Problems
Try each one yourself first. Then tap to check your full solution.
Easy
Express 2 × 2 × a × a in exponential form, and write the numerical value of 2 × 10³.
First part: 2 appears twice and a appears twice, so 2 × 2 × a × a = 2² × a².
Second part: 10³ = 1000, so 2 × 10³ = 2 × 1000 = 2000.
Express 648 as a product of powers of its prime factors in exponential form.
Divide by 2 repeatedly: 648 = 2 × 324 = 2 × 2 × 162 = 2 × 2 × 2 × 81. So three 2s, and 81 is left.
Factor 81 by 3: 81 = 3 × 3 × 3 × 3 — four 3s.
So 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2³ × 3⁴.
Check: 8 × 81 = 648. Correct.
Simplify 10⁻² × 10⁻⁵ and write the answer in exponential form.
Same base (10), multiplying — so add the exponents: (−2) + (−5) = −7.
So 10⁻² × 10⁻⁵ = 10⁻⁷ (which is the same as 1/10⁷).
Medium
Simplify the product m⁵ × n¹² × (m × n)⁹ and write the answer in exponential form.
First open the bracket using the same-power rule: (m × n)⁹ = m⁹ × n⁹.
So the whole product becomes m⁵ × n¹² × m⁹ × n⁹.
Now gather each base and add its exponents (product law, same base):
- For m: m⁵ × m⁹ = m⁵⁺⁹ = m¹⁴.
- For n: n¹² × n⁹ = n¹²⁺⁹ = n²¹.
So the simplified form is m¹⁴n²¹.
If 12² = 144, find (1.2)², (0.12)² and 120² without a fresh multiplication.
The digits are the same as 12; only the decimal point or zeros shift. Squaring shifts the decimal places by double.
- (1.2)²: 1.2 has 1 decimal place, so the square has 2 decimal places: 1.44.
- (0.12)²: 0.12 has 2 decimal places, so the square has 4: 0.0144.
- 120²: 120 = 12 × 10, so 120² = 12² × 10² = 144 × 100 = 14400.
Identify the greater number in each pair: (i) 4³ or 3⁴ (ii) 2⁸ or 8² (iii) 100² or 2¹⁰⁰.
Work out or compare each.
(i) 4³ = 64 and 3⁴ = 81. So 3⁴ is greater.
(ii) 2⁸ = 256 and 8² = 64. So 2⁸ is greater.
(iii) 100² = 10,000 (a 5-digit number). 2¹⁰⁰ is astronomically large — already 2¹⁰ = 1024 is bigger than 100², and 2¹⁰⁰ is far beyond. So 2¹⁰⁰ is greater.
Challenge
64 is both a square (8²) and a cube (4³). Are there other numbers that are both perfect squares and perfect cubes? Describe all such numbers.
Yes — there are infinitely many.
A number that is both a square and a cube must be a 6th power, because 6 is the smallest number divisible by both 2 (for squares) and 3 (for cubes).
Why? If a number is n⁶, then n⁶ = (n³)² (so it is a square) and n⁶ = (n²)³ (so it is a cube). Both at once.
So the numbers that are both squares and cubes are exactly the 6th powers:
- 1⁶ = 1
- 2⁶ = 64
- 3⁶ = 729
- 4⁶ = 4096, and so on.
In general, every number of the form n⁶ (for a whole number n) is both a perfect square and a perfect cube. There are infinitely many.
A dairy makes 8.5 billion milk packets a year and wants a unique numeric ID for each, using only the digits 0–9. How many digits must each ID code have?
First, how many codes can n digits give? Each of the n positions has 10 choices (0–9), so the count of possible codes is 10ⁿ.
We need at least 8.5 billion codes. Write that out: 8.5 billion = 85,00,00,00,000 = 8,500,000,000.
Compare with powers of 10:
- 10⁹ = 1,000,000,000 (one billion) — too few, since 8.5 billion is bigger.
- 10¹⁰ = 10,000,000,000 (ten billion) — enough, since 10 billion is more than 8.5 billion.
So 9 digits are too few but 10 digits are enough. Each ID code must have at least 10 digits.
The world has about 10⁹ sheep and about 10⁹ goats. What is the total population of sheep and goats together, as a single expression?
This is a classic trap. We are adding, not multiplying.
10⁹ + 10⁹ = 2 × 10⁹ (two lots of 10⁹).
It does not become 10¹⁸ (that would be multiplying), and it does not become 2¹⁸ either. When the powers are the same, you just have two copies, so the answer is 2 × 10⁹ (which is 2 billion).
This is exactly why exponent laws are about multiplication and division — there is no shortcut that simply adds the exponents when you add the numbers.
Summary
- An exponent is a shortcut for repeated multiplication. In nᵃ, the base n is multiplied by itself a times, and the exponent counts how many times.
- Product law: aᵐ × aⁿ = aᵐ⁺ⁿ (same base — add the exponents), because you are just counting all the factors together.
- Quotient law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, a ≠ 0 — subtract), because matching factors cancel.
- Power of a power: (aᵐ)ᵇ = aᵐ×ᵇ (multiply the exponents).
- Same exponent: aⁿ × bⁿ = (a × b)ⁿ and aⁿ ÷ bⁿ = (a ÷ b)ⁿ.
- Zero exponent: a⁰ = 1 (a ≠ 0), chosen so the division law stays consistent.
- Negative exponent: a⁻ⁿ = 1/aⁿ — a negative power means reciprocal, never a negative number.
- Powers of 10 give place value: each place left is ×10, each place right is ÷10 (so 10⁰ = 1 sits at the ones place).
- Scientific (standard) form writes a number as x × 10ʸ with 1 ≤ x < 10 and y an integer. The exponent y tells you the size at a glance — check it before the coefficient.
What’s Next
You can now tame both giants and midgets — writing the Earth’s mass or an atom’s size in a few clean symbols, and combining them with the laws of exponents.
Next, in Chapter 3 — A Story of Numbers, you will travel through how number systems grew over history — how people across the world invented ways to write and work with numbers. The powers-of-10 thinking you built here is the very heart of how our number system works. Onward!
Frequently Asked Questions
What is an exponent and what does the base mean?
In a power like 5 to the power 4, the base is 5 (the number you multiply) and the exponent is 4 (how many times you multiply it). So 5 to the power 4 means 5 times 5 times 5 times 5, which equals 625. The exponent is a short way to write repeated multiplication.
What are the main laws of exponents in Class 8?
For the same base a: a to the m times a to the n equals a to the (m plus n); a to the m divided by a to the n equals a to the (m minus n); and (a to the m) to the n equals a to the (m times n). Also a to the power 0 equals 1, and a to the power minus n equals 1 over a to the n. For the same power: a to the n times b to the n equals (a times b) to the n.
Why is any number to the power zero equal to 1?
Use the division law. a to the m divided by a to the m must equal a to the (m minus m), which is a to the power 0. But any number divided by itself is 1. So a to the power 0 has to equal 1, as long as a is not 0. It is a rule chosen so the laws of exponents stay consistent.
What does a negative exponent mean?
A negative exponent means take the reciprocal, that is flip the number to the bottom. For example 2 to the power minus 3 equals 1 over 2 to the power 3, which is 1 over 8. It comes from the division law: when you keep halving past 1, the exponent goes below zero. A negative exponent never makes a number negative.
What is scientific notation or standard form?
Scientific notation writes a number as x times 10 to the power y, where x is at least 1 and less than 10, and y is any integer. For example 5900 becomes 5.9 times 10 to the power 3, and 0.00046 becomes 4.6 times 10 to the power minus 4. The exponent tells you how big or small the number is at a glance.