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2003-A: Powers of Ten

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2003-A: Powers of Ten

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Scientific notation is based on multiplying values but the powers of ten. In order to use scientific notation correctly, we need a thorough understanding of the powers of ten.

Positive Powers of Ten

Any number raised to a positive exponent is that number multiplied by iteself as many times as the exponent. For example $$ 5^4 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 $$

10 raised to any positive power is 10 times itself that many times. A very quick way to do this is to simply add as many zeroes as the value of the exponent: $$ 10^4 = 10000 $$ $$ 10^6 = 1000000 $$

Any number raised to the power of 1 is equal to that number. Thus, \( 10^1 = 10 \).

Problems:

Evalute each power of ten

  1. \( 10^5 = \)
  2. \( 10^3 = \)
  3. \( 10^1 = \)
Negative Exponents

Any number raised to a negative exponent is equal to 1 divided by the number to the positive power. This is called the recipricol. $$ 5^{-6} = \frac{1}{5^5} = \frac{1}{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5} $$

Ten raised to a negative exponent is equal to one divided by 10 raised to a posititive power. This is equal to a decimal value of several zeroes followed by 1. $$ 10^{-5} = \frac{1}{10^5} = \frac{1}{100000} = 0.00001 $$ The number of zeroes between the decimal and the 1 is equal to the value of the exponent minus 1. For example, \( 10^{-5} = 0.00001 \) has 4 zeroes between the decimal and the 1, and \( 10^{-7} = 0.0000001 \) has 6 zeroes between the decimal and the one.

$$ 10^{-1} = \frac{1}{10} = 0.1 $$

Problems:

Evaulate each power of ten as a decimal value.

  1. \( 10^{-4} = \)
  2. \( 10^{-2} = \)
  3. \( 10^{-8} = \)
  4. \( 10^{-1} = \)
Powers of zero:

Any value raised to the power of zero is equal to 1. $$ 5^0 = 1 $$ Thus, $$ 10^0 = 1 $$

  1. \( 8^0 = \)
  2. \( 12345678.98765^0 = \)
  3. \( 10^0 = \)
The Full Spectrum:

Fill out the table below, comprising a full range of powers of ten

Answers:

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