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Powers

Numbers · level 5, lesson 3 of 7

A question to start

Take a sheet of paper and fold it in half. It is now 2 layers thick. Fold again: 4 layers. Again: 8 layers. Each fold doubles the thickness. Ordinary paper is about 0.1 mm thick. If you could fold it 42 times (you cannot: real paper gives up after about seven folds), how thick would it be?

Make a guess: a metre? The height of a house? The answer is about 440 000 km, further than the Moon, which is about 384 000 km away. Forty-two doublings make a number of layers that is over four trillion. Repeated multiplying grows astonishingly fast, and it needs its own short way of writing. That is what powers are.

The idea

Powers: a short way to write repeated multiplying

Multiplication is a short way to write repeated adding: \(5 + 5 + 5 = 3 \times 5\). In the same way, a power is a short way to write repeated multiplying:

\[5 \times 5 \times 5 = 5^3\]

The big number at the bottom, 5, is the base: the number being multiplied. The small raised number, 3, is the exponent (also called the index or the power): how many copies of the base are multiplied together. We read \(5^3\) as "five to the power three", or "five cubed".

\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \qquad 10^4 = 10 \times 10 \times 10 \times 10 = 10\,000\]

\(a^1\) is just \(a\) (one copy, nothing to multiply).

Squares and cubes as pictures

The power 2 and the power 3 have special names, and both come from shapes.

3² = 3 × 3 = 9 2³ = 2 × 2 × 2 = 8
  • \(3^2\) ("three squared") is the number of unit squares in a square 3 units on each side: 9.
  • \(2^3\) ("two cubed") is the number of unit cubes in a cube 2 units along each edge: 8.

The first few squares are \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100\) and the first few cubes are \(1, 8, 27, 64, 125\). (Notice 64 is in both lists: \(8^2 = 4^3 = 64\).) Powers beyond 3 have no picture in our three-dimensional world, but they make perfect sense as repeated multiplying.

Powers of 10

Powers of 10 are the backbone of place value:

\(10^1\) \(10^2\) \(10^3\) \(10^4\) \(10^5\) \(10^6\) \(10^9\)
10 100 1000 10 000 100 000 1 000 000 (a million) 1 000 000 000 (a billion)

The exponent counts the zeros after the 1. Each place in a number is worth a power of 10: in 3 507, the 3 means \(3 \times 10^3\) and the 5 means \(5 \times 10^2\). You will use this a lot in scientific notation.

Negative bases and brackets

A negative base works with the sign rules from lesson 1: each factor is one flip.

\[(-2)^3 = (-2) \times (-2) \times (-2) = -8 \qquad (-2)^4 = 16\]

An odd power of a negative number is negative; an even power is positive.

Be careful with brackets. By the order of operations (level 4), powers come before the minus sign in front:

\[-3^2 = -(3 \times 3) = -9 \qquad \text{but} \qquad (-3)^2 = (-3) \times (-3) = 9\]

Fractions work too: \(\left(\tfrac23\right)^3 = \tfrac23 \times \tfrac23 \times \tfrac23 = \tfrac{8}{27}\).

The laws of exponents: write out the factors

There are a few short rules for working with powers. You do not need to memorise them as magic, because each one explains itself the moment you write out the factors. In all of them, \(a\) is any number and \(m\), \(n\) are positive whole numbers.

Law 1. Multiplying powers of the same base: add the exponents.

\[2^3 \times 2^4 = \underbrace{(2 \times 2 \times 2)}_{3} \times \underbrace{(2 \times 2 \times 2 \times 2)}_{4} = 2^7\]

Three 2s and four 2s make seven 2s. In general, \(m\) copies and \(n\) copies make \(m + n\) copies:

\[a^m \times a^n = a^{m+n}\]

Law 2. Dividing powers of the same base: subtract the exponents.

\[\frac{2^6}{2^2} = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2} = 2^4\]

Each 2 on the bottom cancels one 2 on top, leaving \(6 - 2 = 4\) of them. In general (for now, with \(m\) bigger than \(n\) and \(a\) not 0):

\[\frac{a^m}{a^n} = a^{m-n}\]

Law 3. A power of a power: multiply the exponents.

\[(2^3)^4 = 2^3 \times 2^3 \times 2^3 \times 2^3 = 2^{3+3+3+3} = 2^{12}\]

Four groups of three 2s is \(4 \times 3 = 12\) twos. In general

\[(a^m)^n = a^{mn}\]

Law 4. A power of a product: give the power to each factor.

\[(2 \times 5)^3 = (2 \times 5)(2 \times 5)(2 \times 5) = (2 \times 2 \times 2)(5 \times 5 \times 5) = 2^3 \times 5^3\]

We can regroup like this because the order of multiplying does not matter. In general \((ab)^n = a^n b^n\), and in the same way for a fraction, \(\left(\tfrac{a}{b}\right)^n = \tfrac{a^n}{b^n}\).

What the laws do NOT say

The laws are about multiplying and dividing. There are no such laws for adding:

  • \(2^3 + 2^4 = 8 + 16 = 24\), which is not \(2^7 = 128\).
  • \((2 + 3)^2 = 5^2 = 25\), which is not \(2^2 + 3^2 = 13\).

And Law 1 needs the same base: \(2^3 \times 3^2 = 8 \times 9 = 72\), and there is no way to write it as one power.

Worked examples

1. Evaluate \(5^3\), \((-2)^5\) and \(\left(\tfrac34\right)^2\).

  1. \(5^3 = 5 \times 5 \times 5 = 125\).
  2. \((-2)^5\): the size is \(2^5 = 32\), and an odd power of a negative is negative, so \(-32\).
  3. \(\left(\tfrac34\right)^2 = \tfrac{3^2}{4^2} = \tfrac{9}{16}\).

2. Use the laws. Write each as a single power, then find its value.

  1. \(3^4 \times 3^5 = 3^{4+5} = 3^9 = 19\,683\).
  2. \(\dfrac{7^8}{7^5} = 7^{8-5} = 7^3 = 343\).
  3. \((2^3)^4 = 2^{3 \times 4} = 2^{12} = 4096\).

3. Change the base. Write \(4^3 \times 8^2\) as a power of 2.

  1. \(4 = 2^2\) and \(8 = 2^3\).
  2. So \(4^3 = (2^2)^3 = 2^6\) and \(8^2 = (2^3)^2 = 2^6\) (Law 3).
  3. \(2^6 \times 2^6 = 2^{12} = 4096\) (Law 1).

4. A clever product. Work out \(2^5 \times 5^5\) without a calculator.

By Law 4 backwards, \(2^5 \times 5^5 = (2 \times 5)^5 = 10^5 = 100\,000\). Pairing each 2 with a 5 makes a 10.

5. Which is bigger, \(2^{30}\) or a billion (\(10^9\))?

  1. \(2^{10} = 1024\), a little more than \(1000 = 10^3\).
  2. \(2^{30} = (2^{10})^3 = 1024^3\) (Law 3), and \(1000^3 = 10^9\).
  3. Since \(1024 > 1000\), \(2^{30}\) is bigger. (In fact \(2^{30} = 1\,073\,741\,824\).) The fact that \(2^{10}\) is close to 1000 is why computer memory sizes like "kilo" and "mega" are slightly bigger than a thousand and a million.

Try it

  1. Evaluate: (a) \(3^4\) (b) \((-5)^3\) (c) \(\left(\tfrac12\right)^5\) (d) \(-2^4\)
  2. Write as a single power: (a) \(5^3 \times 5^6\) (b) \(a^4 \times a^7\) (c) \(10^2 \times 10^3 \times 10\)
  3. Work out \(\dfrac{6^9}{6^7}\) by first writing it as a single power.
  4. Work out \((3^2)^3\). Is it the same as \(3^{2^3}\), which means \(3^8\)?
  5. Work out \(4^3 \times 25^3\) without a calculator.
  6. Write \(9^4\) as a power of 3, and \(27^2\) as a power of 3.
  7. Which is bigger, \(3^{20}\) or \(2^{30}\)? Explain without working them out. (Hint: write both as tenth powers.)
  8. What is the last digit of \(2^{100}\)? (Hint: look at the last digits of \(2^1, 2^2, 2^3, \ldots\))
Answers
  1. (a) \(81\). (b) \(-125\) (odd power of a negative). (c) \(\tfrac{1}{32}\). (d) \(-16\): the power comes first, so it is \(-(2^4)\).
  2. (a) \(5^9\). (b) \(a^{11}\). (c) \(10 = 10^1\), so \(10^{2+3+1} = 10^6\).
  3. \(6^{9-7} = 6^2 = 36\).
  4. \((3^2)^3 = 3^6 = 729\). It is not the same as \(3^8 = 6561\). A power of a power multiplies the exponents (\(2 \times 3 = 6\)); a "tower" \(3^{2^3}\) means 3 to the power \(2^3 = 8\).
  5. \(4^3 \times 25^3 = (4 \times 25)^3 = 100^3 = 1\,000\,000\).
  6. \(9 = 3^2\), so \(9^4 = (3^2)^4 = 3^8\). \(27 = 3^3\), so \(27^2 = 3^6\).
  7. \(3^{20} = (3^2)^{10} = 9^{10}\) and \(2^{30} = (2^3)^{10} = 8^{10}\). Ten 9s multiplied make more than ten 8s multiplied, so \(3^{20}\) is bigger. (Values: \(3^{20} = 3\,486\,784\,401\) and \(2^{30} = 1\,073\,741\,824\).)
  8. The last digits go \(2, 4, 8, 6, 2, 4, 8, 6, \ldots\), repeating every 4, because the last digit of a product only depends on the last digits being multiplied. \(2^4, 2^8, 2^{12}, \ldots\) all end in 6. Since 100 is a multiple of 4, \(2^{100}\) ends in 6.

Common mistake

Multiplying the base by the exponent. \(2^3\) is \(2 \times 2 \times 2 = 8\), not \(2 \times 3 = 6\). The exponent counts how many copies to multiply; it is not a number to multiply by.

Using the laws on sums. \(2^3 + 2^4 \ne 2^7\), and \((a + b)^2 \ne a^2 + b^2\). Write out the factors if you are not sure: the laws only come from multiplying.

Where it leads