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  1. 1. Power, Roots, Logarithms
  2. 2. Power • In mathematics, that which is represented by an exponent or index, denoted by a superior numeral. A number or symbol raised to the power of 2 – that is, multiplied by itself – is said to be squared (for example, 32, x2), and when raised to the power of 3, it is said to be cubed (for example, 23, y3). Any number to the power zero always equals 1. Powers can be negative. Negative powers produce fractions, with the numerator as one, as a number is divided by itself, rather than being multiplied by itself, so for example 2-1 = 1/2 and 3-3 = 1/27.
  3. 3. Product Rule The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut! Power Rule The "power rule" tells us that to raise a power to a power, just multiply the exponents. Here you see that 52 raised to the 3rd power is equal to 56.
  4. 4. Quotient Rule The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown. Zero Rule According to the "zero rule," any nonzero number raised to the power of zero equals 1. Negative Exponents The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.
  5. 5. Summary of index laws
  6. 6. Roots and Radicals We use the radical sign : It means "square root". The square root is actually a fractional index and is equivalent to raising a number to the power 1/2. So, for example: 251/2 = √25 = 5 You can also have Cube root: (which is equivalent to raising to the power 1/3), and Fourth root: (power 1/4) and so on.
  7. 7. Things to remember If a ≥ 0 and b ≥ 0, we have: However, this only works for multiplying. Please note that does not equal Also, this one is often found in mathematics:
  8. 8. Logarithms Two kinds of logarithms are often used : common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number. The power to which the base e (e = 2.718281828.......) must be raised to obtain a number is called the natural logarithm (ln) of the number. Log Rules: 1) logb(mn) = logb(m) + logb(n) 2) logb(m/n) = logb(m) – logb(n) 3) logb(mn) = n · logb(m) 4) logb = logb x1/y = (1/y )logb x