10.1.5 Negative Exponents

Notepad iconActivity: Further Investigation of Exponents

Work out 10 squared division 10 super six the “long way” and using the rules for powers with the same base.

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Answer

equation sequence 10 squared division 10 super six equals 10 squared divided by 10 super six equals 10 multiplication 10 divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 super four

The rule gives equation sequence 10 squared division 10 super six equals 10 super open two minus six close equals 10 super four minus, so 10 super four minus means one divided by 10 super four.

We read 10 super four minus as “10 to the power negative four.” [ You learned about reciprocals in Unit 7 when dividing fractions. ] Often, the word “power” is left out and we say “10 to the negative four” instead. In the same way, five super two minus is said as “five to the negative two” and means one divided by five squared, or one divided by five multiplication five, which is equivalent to one divided by 25.

Alternatively, you can say that five super two minus is the reciprocal of five squared.

Activity Symbol Activity: Understanding Negative Powers

Without using your calculator, write the following numbers as fractions or whole numbers, as appropriate.

(a) two super three minus

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Answer

(a) equation sequence two super three minus equals one divided by two cubed equals one divided by two multiplication two multiplication two equals one divided by eight

(b) three super four minus

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Answer

(b) equation sequence three super four minus equals one divided by three super four equals one divided by three multiplication three multiplication three multiplication three equals one divided by 81

(c) six super zero

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Comment

What does any number to the power 0 equal?

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Answer

(c) six super zero equals one

(d) 10 super six minus

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Answer

(d) equation sequence 10 super six minus equals one divided by 10 super six equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by one comma 000 comma 000

Notepad iconActivity: Writing Negative Numbers

Write the following as powers of 10.

(a) 0.01

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Comment

Convert the number to a fraction first, thinking back to place value if you need to.

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Answer

(a) equation sequence 0.01 equals one divided by 100 equals one divided by 10 squared equals 10 super two minus

(b) 0.0000001

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Answer

(b) equation sequence 0.0000001 equals one divided by 10 comma 000 comma 000 equals one divided by 10 super seven equals 10 super seven minus

10.1.4 Zero as a Power

10.1.6 Small numbers and Scientific Notation