The discovery of exponents gave us the capability to communicate mathematical ideas in an efficient manner that was previously impossible. In addition, the use of this new form of notation led to the discovery of rules pertaining to the use of exponents. We will examine these rules in upcoming lessons.

The exponent of a number indicates how many times to use
that number under multiplication.
 
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The word "exponent" is often synonymous with the word "power".
62 can be read as "6 raised to a power of 2" or "6 squared".

The use of an exponent is referred to as repeated multiplication.
(Remember that multiplication is referred to as repeated addition.)

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Examples: • 34 = 3 × 3 × 3 × 3
• 127 = 12 × 12 × 12 × 12 × 12 × 12 × 12

• 41 = 4
  • 60 = 1     (any number raised to a power of 0 is one, but 00 is undefined)
• 53 = 5^3     (alternate notation often seen on computers and calculators)
• an = a × a × a × . . . × a       
(n multiples of the value of a)

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Exponents of Negative Values:

When multiplying negative numbers together,
parentheses will be needed when switching to exponential notation. Consider:
(-4)(-4)(-4)(-4)(-4)(-4) = (-4)6

Note: The examples below show that
(-4)6 and -46 are not the same.

Examples:
(negative values)
(-3)2 = (-3) × (-3) = +9  The 2 is "attached" to the parentheses, so everything inside
the parentheses is squared.
  
beware
-32 is not the same as (-3)2
   -32 = -(32)       The 2 is "attached" to the 3, but not to the negative sign.
   -32 = -(32) = -(3 × 3) = -(9) = -9
The expression -32 (with missing parentheses) means to multiply two 3's together first (by order of operations), and then take the negative of that answer.

Even powers of negative numbers allow for the negative values to be arranged in pairs. This pairing guarantees that the answer will always be positive. Remember, a negative number times another negative number yields a positive result.

(-3)6 = (-3)(-3) • (-3)(-3) • (-3)(-3)
= 9 • 9 • 9
= 729
(positive)

Odd powers of negative numbers, however, always leave one factor of the negative number not paired. This one lone negative term guarantees that the answer will always be negative.

(-3)5 = (-3)(-3) • (-3)(-3) • (-3) ← lone factor
= 9 • 9 • (-3)
= -81
(negative)

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Exponents and Units:

When working with units and exponents (or powers),
remember to adjust the units appropriately:

(25 in)3    
= (25 in) • (25 in) • (25 in)
= (25 • 25 • 25 )(in • in • in)
= 15625 in3

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ti84c
For calculator help with exponents
click here.

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