How Powers, Roots and Scientific Notation Work — A Complete Guide

A square with sides of 12 centimetres has an area of 144 square centimetres. A quantity that doubles ten times grows from 1 to 1,024. The speed of light is about 300,000,000 metres per second, a number so long that writing it out in full is an invitation to miscount the zeros. All three of these are questions about powers: a short way of writing repeated multiplication, and a way of undoing it, and a way of writing very large and very small numbers so that they stay readable. Powers look compact and a little mysterious at first, but they follow a small set of rules, and once those rules are clear, the whole topic becomes a matter of careful bookkeeping.

This guide explains what a power means, how squares and square roots undo each other, what cubes and cube roots do, the laws for multiplying and dividing powers, what zero and negative exponents mean, and how scientific notation uses powers of ten to handle very large and very small numbers. It works through three examples — the area of a square, a quantity that keeps doubling, and a multiplication written in scientific notation — and it shows where people most often go wrong, so you can check your own answers with confidence.

What a power is

A power is repeated multiplication written once. The expression 25 means 2 multiplied by itself five times: 2 × 2 × 2 × 2 × 2, which is 32. The larger number, 2, is called the base, and the smaller raised number, 5, is called the exponent or index. The exponent says how many copies of the base are multiplied together. So 34 is 3 × 3 × 3 × 3, which is 81, and 53 is 5 × 5 × 5, which is 125. Reading a power aloud helps: 25 is “two to the power of five”, 34 is “three to the power of four”, and 104 is “ten to the power of four”, which is 10,000.

The point of the notation is that repeated multiplication grows very fast, and writing it out in full quickly becomes impractical. Ten copies of 2 multiplied together is 210, which is 1,024, and writing ten twos with nine multiplication signs is already clumsy. Twenty copies would be unusable. Powers also make the pattern visible: each extra multiplication by the base multiplies the result by the base again, so 25 is 32, 26 is 64, and 27 is 128. That steady multiplying is exactly how doubling works, and it is why powers appear wherever something grows by repeated multiplication, from interest building on a sum of money to a population doubling over time.

Two special cases of wording are worth learning early. A number to the power of 2 is called squared, because it gives the area of a square with that side length, and a number to the power of 3 is called cubed, because it gives the volume of a cube with that edge length. The exponent 1 needs no work at all: any number to the power of 1 is itself, so 71 is 7. And a power is not the same as a multiplication: 23 is 8, not 6, because the exponent counts multiplications, not a multiplication by the exponent. That single confusion — reading 23 as 2 × 3 — is the most common error in the whole topic, and it is worth checking for every time a power is evaluated.

Squares and square roots

Squaring a number means multiplying it by itself: 122 is 12 × 12, which is 144, and 152 is 15 × 15, which is 225. The square root undoes squaring: it asks which number, multiplied by itself, gives the number you started with. The square root of 144 is 12, because 12 × 12 is 144, and the square root of 225 is 15. The symbol for a square root is √, so √144 = 12 and √225 = 15. Squaring and taking a square root are inverse operations, in the same way that adding and subtracting undo each other: one builds the square, the other recovers the side.

Not every number has a whole-number square root. The square root of 2 is about 1.414, because 1.414 × 1.414 is very close to 2, but no fraction or terminating decimal multiplied by itself gives exactly 2. Numbers like this, whose decimal expansion goes on without repeating, are called irrational numbers, and in practice they are left as √2 or rounded to a sensible number of decimal places. A useful check for any square root is to square the answer: if √50 is about 7.071, then 7.071 × 7.071 should come back to about 50, and it does. If the check lands far from the starting number, the root is wrong.

Square roots appear whenever a squared quantity has to be undone. If a square garden has an area of 100 square metres, each side is √100, which is 10 metres. If a square has an area of 144 square centimetres, each side is 12 centimetres. The same idea works in reverse for checking: a side of 12 centimetres squared gives the area of 144 square centimetres, so the two statements agree. Keeping the pair together — square one way, root the other — makes it much harder to mix up which operation a problem is asking for.

Cubes and cube roots

Cubing a number means multiplying it by itself three times: 33 is 3 × 3 × 3, which is 27, and 43 is 4 × 4 × 4, which is 64. The cube root undoes cubing: the cube root of 27 is 3, because 3 × 3 × 3 is 27, and the cube root of 64 is 4. The symbol is ∛, so ∛27 = 3 and ∛64 = 4. Just as a square gives the area of a square, a cube gives the volume of a cube: a box with edges of 4 centimetres holds 64 cubic centimetres, and recovering the edge from the volume means taking the cube root.

Cubes grow faster than squares, which is why the distinction matters in practice. Doubling the side of a square multiplies its area by 4, because 22 is 4, but doubling the edge of a cube multiplies its volume by 8, because 23 is 8. A box that is twice as wide, twice as deep and twice as tall holds eight times as much, not twice as much. That is the same repeated-multiplication idea as before, with one more copy of the base, and it is a good test of whether the exponent has been understood: the exponent counts the copies, and each copy multiplies the result again.

The laws of indices

Powers with the same base can be combined without writing out all the multiplications, and the rules for doing so are called the laws of indices. To multiply powers with the same base, add the exponents: 23 × 24 is 27, which is 128, because three copies of 2 and then four more copies make seven copies in total. To divide powers with the same base, subtract the exponents: 25 ÷ 22 is 23, which is 8, because five copies with two cancelled leave three. And a power of a power multiplies the exponents: (23)2 is 26, which is 64, because two groups of three copies make six copies.

Each law can be checked by writing the multiplications out once, and doing so is the best way to remember them. For (32)3, write 32 three times — (3 × 3) × (3 × 3) × (3 × 3) — and count six threes, giving 36, which is 729. The laws only work when the bases match: 23 × 32 cannot be combined into a single power, because the copies are of different numbers, and it must be evaluated as 8 × 9, which is 72. Likewise, a power of a sum cannot be split apart: (3 + 4)2 is 72, which is 49, while 32 + 42 is 9 + 16, which is 25, and the two are not equal. The exponent applies to the whole bracket, not to each term inside it.

The order of operations still governs everything else. Powers are evaluated before multiplication and division, and brackets come first, as in the order of operations guide. So 2 × 32 is 2 × 9, which is 18, not 62, and (2 × 3)2 is 62, which is 36. A negative base needs the same care: (−2)3 is −8, because three copies of −2 multiply to a negative, while (−2)2 is 4, because two negatives make a positive. But −22, without brackets, means the negative of 22, which is −4, because the power is evaluated before the minus sign. Brackets decide which reading is meant, and leaving them out is a frequent source of sign errors.

Zero and negative exponents

Any non-zero number to the power of 0 is 1: 50 is 1, and so is 1000. This is not an arbitrary rule; it follows from the division law. Dividing 23 by 23 gives 1, because any non-zero number divided by itself is 1, and subtracting the exponents gives 20, so 20 must be 1 for the law to stay consistent. The same pattern appears in the sequence 23 = 8, 22 = 4, 21 = 2, where each step divides by 2, and the next step gives 20 = 1. Zero as an exponent means no copies are multiplied, and the empty product is 1, in the same way that starting with 1 and multiplying by nothing leaves 1.

A negative exponent means a reciprocal: 2−3 is 1 ÷ 23, which is 1/8, or 0.125, and 10−2 is 1 ÷ 102, which is 1/100, or 0.01. Continuing the halving sequence past 1 gives 2−1 = 1/2, 2−2 = 1/4, and 2−3 = 1/8, which matches the subtraction law: 20 ÷ 23 is 2−3, and 1 ÷ 8 is 1/8. Negative exponents are therefore not negative answers: 2−3 is a small positive number, not −8, and the minus sign belongs to the exponent, not to the result. The only exponent that is genuinely special is 00, which is left undefined in school mathematics, because the rules for 0 and for exponents point in different directions there; it does not arise in ordinary calculation.

Scientific notation

Scientific notation writes a number as a number between 1 and 10 multiplied by a power of ten. The number 300,000,000 becomes 3 × 108, because the decimal point has moved eight places, and 6,400,000 becomes 6.4 × 106. Small numbers work the same way with negative exponents: 0.000025 becomes 2.5 × 10−5, because the decimal point has moved five places to reach 2.5. The first number, called the coefficient, must be at least 1 and less than 10, and the exponent counts how many places the decimal point moved. Moving the point to the right, for a number larger than 10, gives a positive exponent; moving it to the left, for a number smaller than 1, gives a negative exponent.

The notation earns its keep when numbers are multiplied or divided. To multiply in scientific notation, multiply the coefficients and add the exponents: (2 × 103) × (3 × 104) is 6 × 107, which is 60,000,000, because 2,000 × 30,000 is 60,000,000. To divide, divide the coefficients and subtract the exponents: (4.8 × 106) ÷ (1.2 × 103) is 4 × 103, which is 4,000. Adding and subtracting in scientific notation is different: the exponents must match first, because the numbers must be in the same units before they can be combined, just as metres and kilometres must be converted before adding. So 3 × 105 + 2 × 105 is 5 × 105, which is 500,000, but 3 × 105 + 2 × 104 must first be rewritten, for example as 3 × 105 + 0.2 × 105, giving 3.2 × 105, which is 320,000.

Scientific notation is the standard way of writing measurements in science, because it keeps the important digits visible and the size of the number in the exponent. A result written as 6.4 × 106 shows two significant figures and a size of millions at a glance, while 6,400,000 hides both behind a string of zeros. The same notation appears on the scientific calculator on this site, which can evaluate powers and roots directly; using it to check a hand calculation is good practice, but the notation itself has to be read correctly first, or the calculator's answer will be misread by a factor of ten. The most common misreading is treating the exponent as a count of zeros to append after the coefficient: 2.5 × 104 is 25,000, not 2,500, because the exponent counts places moved from 2.5, and 2.5 × 10,000 is 25,000.

Worked example 1: the area of a square

A square has sides of 12 centimetres. Its area is 122, which is 12 × 12, or 144 square centimetres. Now suppose each side is doubled, to 24 centimetres. The new area is 242, which is 576 square centimetres, and 576 divided by 144 is 4, so doubling the side has multiplied the area by 4. That factor is 22, and it follows directly from the power: (2 × 12)2 is 22 × 122, which is 4 × 144. The check works in reverse too: the square root of 576 is 24, because 24 × 24 is 576, so the side has been recovered from the area. Whenever a length is scaled, the area scales by the square of the factor, and that is why the exponent 2 appears in every area formula.

Worked example 2: a quantity that keeps doubling

A quantity starts at 1 and doubles at each step. After 1 step it is 2, after 2 steps it is 4, and after 10 steps it is 210, which is 1,024. Writing the steps out shows the pattern: 25 is 32, 26 is 64, 27 is 128, 28 is 256, 29 is 512, and 210 is 1,024. The laws of indices give a shortcut for a later step: after 10 steps, five more doublings multiply the quantity by 25, so the total after 15 steps is 210 × 25, which is 215, or 32,768. The shortcut is checked by the same rule in reverse: 215 ÷ 210 is 25, which is 32, and 32,768 ÷ 1,024 is indeed 32. Powers turn a long chain of multiplications into one addition of exponents.

Worked example 3: multiplying in scientific notation

Multiply 2 × 103 by 3 × 104. Multiply the coefficients first: 2 × 3 is 6. Then add the exponents, by the multiplication law for powers with the same base: 3 + 4 is 7. The answer is 6 × 107, which is 60,000,000. The ordinary-numbers check confirms it: 2 × 103 is 2,000, 3 × 104 is 30,000, and 2,000 × 30,000 is 60,000,000. For division, the same method gives (4.8 × 106) ÷ (1.2 × 103) = 4 × 103, which is 4,000, because 4.8 ÷ 1.2 is 4 and 6 − 3 is 3. If the coefficient after multiplying is 10 or more, it must be adjusted: 5 × 103 × 4 × 102 is 20 × 105, which is rewritten as 2 × 106, or 2,000,000, so that the coefficient is back between 1 and 10.

Common mistakes to avoid

Frequently asked questions

What is the difference between 23 and 32?

They are different powers with the base and exponent swapped. 23 is 2 × 2 × 2, which is 8, while 32 is 3 × 3, which is 9. Powers are not commutative in this way: swapping the base and the exponent usually changes the value, so the order must be read as written, base first and exponent raised.

Why is any number to the power of 0 equal to 1?

It follows from the division law for indices. Dividing a power by itself gives 1, and subtracting the exponents gives an exponent of 0, so the exponent-0 power must be 1 for the law to stay consistent. The sequence 8, 4, 2, 1 for 23, 22, 21, 20 shows the same thing: each step divides by the base, and the step after 2 is 1. The one exception is 00, which is left undefined.

Can a square root be a decimal that never ends?

Yes. The square root of 2 is about 1.41421356…, and its decimal expansion goes on without repeating, because no fraction multiplied by itself gives exactly 2. In practice, such roots are left as √2 in exact working, or rounded to a sensible number of places, and the answer is checked by squaring the rounded value to see that it comes back close to the starting number.

How do I convert an ordinary number into scientific notation?

Move the decimal point until exactly one non-zero digit stands before it, and count the places moved. For 6,400,000, the point moves six places to reach 6.4, giving 6.4 × 106. For 0.000025, the point moves five places to reach 2.5, giving 2.5 × 10−5. The exponent is positive when the original number is larger than 10 and negative when it is smaller than 1, and the coefficient must always be at least 1 and less than 10.