# 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If the €100 is left in the account, and the €5 withdrawn, then each year there would be €5 interest.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.

## Site Index

### Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

### Resources

Science resources on ScienceLibrary.info. Games, puzzles, enigmas, internet resources, science fiction and fact, the weird and the wonderful things about the natural world. Have fun while learning Science with ScienceLibrary.info.

### Great Scientists

#### Archimedes

c. 287 - 212 BCE

Archimedes was a Greek, living in Magna Graecia. He is considered one of the greatest mathematicians and engineers of the ancient world, and the source for many fascinating stories and anecdotes.