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Exponents

$A^B$: A is the base and B is the exponent.

An exponent can also be called power or index. In equations, the highest exponent is called the order of the equation.

Exponents are an essential tool of mathematics. They shorten and make long multiplications and divisions easier. The decimal and metric systems, used in the S.I. system of units, depend on exponents.

Make sure you learn the rules, and are very practised with manipulating exponents, before you try logarithms, which are the inverse of exponents.

Exponential rules

  • $A^n$ = A × A n times.
  • $2^5 = 2×2×2×2×2 = 32$

  • $(A×B)^n = A^n × B^n$
  • $(2×3)^3 = 2^3 × 3^3 = 8 × 27 = 216$

  • $A^B$ × $A^C$ = $A^{B+C}$
  • $2^5 × 2^3 = 2^{5+3} = 2^8 = 256$

  • ${A^B}/{A^C}$ = $A^{B-C}$
  • $6^5 ÷ 6^3 = 6^{5-3} = 6^2 = 36$

  • $({A^B})^C$ = $A^{B×C}$
  • $(1.5^5)^3 = 1.5^{5×3} = 1.5^{15} = 437.9$

  • $A^0 = 1$
  • $3^2 ÷ 3^2 = 3^(2-2) = 3^0 = 1$

  • $A^{1/n}$ = n$√A$
  • $9^{1/3}$ = 3$√{27} = 3$

  • n$√{A^m}$ = $A^{m/n}$
  • 3$√{8^2} = 8^{2/3} = 2^2 = 4$

  • $A^{-n} = 1/{A^n}$
  • $5^{-2} = 1/{5^2} = 1/{25} = 0.04$

Solving Exponential Equations

An exponential function is one where the dependent variable is a power in the equation: e.g. $f(x) = a^x$, where is $a > 0$ and $a≠1$.

Exponentials graphs
Exponentials graphs: all rational and irrational numbers to the power of x pass through (0,1)

Observe that for $a>1$, the curve ascends to the right, and for $a<1$, the curve descends from the left.

A curve with $a>1$ is called an exponential growth function. When $a<1$, the curve is called a exponential decay function.

An example of a growth function is compound interest, in which money invested grows exponentially (there is a positive yield, and the capital increases over time).

An example of a decay function is the half-life of radioactive elements. As time goes on there is less and less of the original substance, but the decrease is not linear, and follows an exponential curve with negative slope.

Law of Indices

Appreciation is the rate of increase in a quantity, and depreciation is the rate of decrease.

Depreciation

A car is bought for €20,000. After 4 years, it is resold for €8,000. Use the law of indices to determine the rate of depreciation.

Let $P_0$ be the initial price, $P_t$ be the price in year $t$, and the rate of depreciation be $r$.

$P_4 = P_0(1-r)^4$ ⇒ $(1-r)^4 = {P_4}/{P_0} = {8000}/{20000} = 0.4$

$r = 1-(0.4)^{1/4} = 0.205$

The car depreciates by an average of 20.5% per annum.

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