Mathematics · Numbers and calculation · Grade 7

Powers: from folding to a formula

How thick is a sheet of paper folded 10 times? How many bacteria will there be by morning?

The situation

An exercise-book page is 0.1 mm thick. Fold it in two — 2 layers, fold again — 4, again — 8. Fold it ten times and you get 1024 layers: thicker than 10 centimetres! Bacteria do the same: every 20 minutes each one splits in two — how many will there be by morning from a single one? Mathematics has a short notation for multiplying the same number again and again — the power. We will discover its rules ourselves with squares.

Topic

Powers with a natural exponent and their properties. Folding paper and a bacterium dividing in two: from repeated multiplication to powers, the laws of indices on a grid

Key ideas

Power
Base and exponent
Properties of powers

Formulas

a^{n} = \underset{n}{\underset{⏟}{a \cdot a \cdot \dots \cdot a}}

A power with a natural exponent: $a$ is the base, $n$ is the exponent (the number of factors). $a^{1} = a$ .

a^{m} \cdot a^{n} = a^{m + n}, a^{m} : a^{n} = a^{m - n} (m > n)

When powers with the same base are multiplied, the exponents are added; when they are divided, the exponents are subtracted. The base does not change!

{(a^{m})}^{n} = a^{m n}, {(a b)}^{n} = a^{n} b^{n}, {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}

A power of a power, a power of a product and a power of a fraction.

a^{0} = 1 (a \neq 0), 0.1 mm \cdot 2^{10} = 102.4 mm

The zero exponent: $a^{m} : a^{m} = 1$ . The thickness of a sheet of paper folded ten times.

Theory

Repeated multiplication — the power

Each time the paper is folded, the layers become 2 times as many: $2 \cdot 2 \cdot 2 \cdot 2 = 16$ . A product of identical factors is written briefly as $2^{4}$ — “two to the power of four”. The repeated number is the base ( $2$ ), and the number of times it is taken is the exponent ( $4$ ). Calculating a power is called raising to a power. Note: $2^{4} = 16$ , but $2 \cdot 4 = 8$ — a power is not a product! With a single factor, $a^{1} = a$ .

Square and cube

The second power is called the square: $a^{2} = a \cdot a$ — the number of cells in a square of side $a$ ( $3^{2} = 9$ ). The third power is the cube: $a^{3} = a \cdot a \cdot a$ — the number of small cubes in a cube of edge $a$ ( $3^{3} = 27$ ): $3$ layers of $9$ each. With a negative base, watch the sign: ${(- 2)}^{3} = - 8$ , ${(- 2)}^{4} = 16$ — with an even exponent the result is positive, with an odd one negative. $- 2^{4} = - (2^{4}) = - 16$ : without brackets, the minus is not part of the power.

Multiplying and dividing

$2^{3} \cdot 2^{2} = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2)$ — $3 + 2 = 5$ factors in all: $2^{5}$ . The rule: when powers with the same base are multiplied, the exponents are added: $a^{m} \cdot a^{n} = a^{m + n}$ . In division the factors cancel in pairs: $2^{5} : 2^{2} = 2^{3}$ , that is, $a^{m} : a^{n} = a^{m - n}$ . If $m = n$ , all the factors cancel and $1$ is left — which is why we take $a^{0} = 1$ (with $a \neq 0$ ). Mistakes: $2^{3} \cdot 2^{2} \neq 4^{5}$ (the bases are not multiplied), and $2^{3} \cdot 3^{2}$ cannot be turned into a single power.

A power of a power, a power of a product

${(2^{3})}^{2} = 2^{3} \cdot 2^{3}$ — $2$ rows with $3$ factors in each, $3 \cdot 2 = 6$ in all: ${(a^{m})}^{n} = a^{m n}$ . If we split a square of side $2 \cdot 3 = 6$ into $2 \times 2$ blocks, each block has $3^{2}$ cells: ${(2 \cdot 3)}^{2} = 2^{2} \cdot 3^{2} = 36$ . In general ${(a b)}^{n} = a^{n} b^{n}$ and ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$ . Powers also make prime factorisations compact: $360 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 = 2^{3} \cdot 3^{2} \cdot 5$ (the “Prime numbers” experiment).

Growth and powers of 10

$2 n$ increases by $2$ at each step, while $2^{n}$ is multiplied by $2$ : at $n = 10$ they are $20$ and $1024$ . That is why folded paper, dividing bacteria and interest in a bank grow “explosively”. Powers of $10$ are the place values: $10^{3} = 1000$ , $10^{6} = 1 000 000$ . Large numbers are written in standard form: $380 000 = 3.8 \cdot 10^{5}$ , where $1 \leq a < 10$ ; the folded paper is $102 400$ µm $= 1.024 \cdot 10^{5}$ µm.

About the model

In this experiment the exponent is a natural number ( $n \geq 1$ ) or zero; negative and fractional exponents are in the “Square and cube roots” experiment. In the drawings the number of cells is limited (at most a few hundred), but the formulae are true for any number. Real paper cannot be folded by hand more than 7–8 times (each fold halves the width), and bacteria multiply only until the food runs out — the $2^{n}$ growth does not go on for ever.

Did you know?

According to legend, the inventor of chess asked for a reward of 1 grain of wheat on the first square of the board, 2 on the second, 4 on the third, and so on. In all that is $2^{64} - 1 = 18 446 744 073 709 551 615$ grains — roughly a thousand years of the whole world's harvest.
Writing the exponent small and raised ( $x^{3}$ ) spread from the book “La Géométrie” (1637) by the French mathematician René Descartes. In 2002 the American schoolgirl Britney Gallivan folded a long strip of paper 12 times and also derived a formula for the number of folds.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Fractions: from pizza to formula — Grade 5
Prime numbers, HCF and LCM — Grade 7
The number line: from −7 to √2 — Grade 7
Roots: from squares to powers — Grade 8