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2003-A: Powers of Ten

• Topic Cluster: Pure Mathematics
• Topic: Scientific Notation and Orders of Magnitude
• Objective: Evaluate positive and negative powers of ten and ten raised to the zeroth power.
• Content:
• Level: 1

BACK to Ladder Scientific Notation and Orders of Magnitude

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