Powers

When 100 is written as $10^2$, we say ten squared. But we can also say ten to the power of two.

When 1000 is written as $10^3$, we say ten cubed. But we can also say ten to the power of three.

If the power is greater than three, we have to use the 'power' to express it: e.g. $4^{4}$ is read 'four to the power of four'. This would be 4 ⋅ 4 ⋅ 4 ⋅ 4 = 256

What about when the number is smaller than two? $10^{1/2}$ is read 'ten to the half', and means the square root of ten ($√{10}$).

What about negative powers? No problem: $10^{-1}$ means $1/{10}$, or 0.1. And $10^{-2} = 1/{10^2} = 1/{100} = 0.01$.

Powers are also called 'exponents'.

power rules

• Zero Exponent Rule

$x^0 = 1$, provided x ≠ 0

• Negative Exponent Rule

$x^{-n} = 1/{x^n}$ and $1/{x^{-n}} = {x^n}$, provided x ≠ 0

• Power of one

$x^1 = x$

• Power of one-half

$x^{1/2} = √x$, when x ≥ 0

• Power of one-third

$x^{1/3} = ∛{x}$. x can be negative.

• Power of one-fourth

$x^{1/4} = {(√x)}^{1/2} = √{√x}$, when x ≥ 0

• Multiplication

$x^a ⋅ x^b = x^{a + b} $

• Division

${x^a}/{x^b} = x^{a - b} $

• Quotient

$({x}/{y})^a = {x^a}/{y^a} $

• Power of a power

${(x^a)}^b = x^{(ab)} $

Interest

Simple Interest

The yield at simple interest = $a + aix$, where $a$ is the initial capital, $i$ the interest rate per annum, and $x$ the number of years.

If €100 is put in a bank at 5% interest, after one year there would be the initial €100 plus €5 (5% of 100 = 5), or €105.

If the €100 is left in the account, and the €5 withdrawn, then each year there would be €5 interest. This is an example of simple interest, where only the capital is earning interest.

Compound Interest

If the interest is left with the initial capital $€100$, then in the second year there is more money (€105) at 5% interest. After two years there would be €105 plus €5.25 (5% of 105 = 5.25), or €110.25.

In a third year, this capital would earn €110.25 + (0.05 ⋅ 110.25) = €110.25 + 5.51 = 115.76.

Each year there is a little more earnings on the interest, so the yield is not linear (the same every year), but exponential.

Compound interest can be written as a formula: Yield = $ab^x$, where a is the initial capital, b is the rate of return (1 plus the interest), and x is the number of years.

In our example, a = 100, b = 1.05, and x = 3: $ab^x = 100 ⋅ (1.05)^3 = 115.76$

Computers and powers of two

Normally, numbers are in base 10. There is no single integer to represent ten, instead we have a position-sensitive system for communicating the values. For example, adding 1 to 9 results in 10.

In base-2, or binary, there are only two integers used, 0 and 1. We count 0, 1, 10, 11, 100, 101, 110, 111, 1000. [$1000_{2}$ = (1 x 8) + (0 x 4) + (0 x 2) + (0 x 1) = 8 in base-10]

Computers use base-2, which explains why memory is given as powers of 2: 256Mb = $2^8$, which in binary can be expressed as 100000000.