10.1.3 Multiplying and Dividing Numbers with the Same Base Number

How does this notation help in working out the width of the universe?

Consider what happens when you multiply two different powers of 10 together, say 10 squared times and 10 super four.

multiline equation line 1 equation left hand side 10 squared multiplication 10 super four equals right hand side open 10 multiplication 10 close multiplication open 10 multiplication 10 multiplication 10 multiplication 10 close line 2 equation sequence equals 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals one comma 000 comma 000

There are six 10s multiplied together, so we can write this as 10 super six. So, 10 super two equation left hand side prefix multiplication of 10 super four equals right hand side 10 super six

Have you noticed anything about the powers of ten in the answer and the original numbers?

You can get the same result by adding the powers together:

10 squared equation sequence prefix multiplication of 10 super four equals 10 super open two plus four close equals 10 super six

Try working out two cubed prefix multiplication of two super five the “long way” and then by adding the powers together. Do your answers agree? The following rule applies in general: To multiply two numbers with the same base number, add the powers.

Activity Symbol Activity: Multiplying Numbers with the Same Base Number

Work out the following, giving your answer first in power form and then in decimal form.

(a) two squared prefix multiplication of two super four

Solution Symbol

Answer

(a) two squared equation sequence prefix multiplication of two super four equals two super open two plus four close equals two super six equals 64

(b) open three minus close squared multiplication open three minus close cubed

Solution Symbol

Answer

(b) equation sequence open three minus close squared multiplication open three minus close cubed equals open three minus close super two plus three equals open three minus close super five equals two minus times 43

(c) 10 super four prefix multiplication of 10 super three

Solution Symbol

Answer

(c) 10 super four equation sequence prefix multiplication of 10 super three equals 10 super open four plus three close equals 10 super seven equals 10 comma 000 comma 000

Notepad iconActivity: The Width of the Universe

You can now work out the width of the universe!

Remember that the universe is 92 billion light years across and a light year is about 9.5 trillion kilometers.

Solutions icon

Answer

Width of the universe = 92 billion prefix multiplication of 9.5 trillion km.

. First express both numbers in scientific notation: 92 billion equals nine .2 prefix multiplication of 10 super 10 and nine .5 trillion equals nine .5 prefix multiplication of 10 super 12.

So the width = 92 billion multiplication nine .5 trillion km equals nine .2 prefix multiplication of 10 super 10 multiplication nine .5 prefix multiplication of 10 super 12 km = 9.2 multiplication nine .5 prefix multiplication of 10 super 10 prefix multiplication of 10 super 12. Multiplying 9.2 by 9.5 gives 87.4. Then adding the powers gives 10 super 10 equation sequence prefix multiplication of 10 super 12 equals 10 super open 10 plus 12 close equals 10 super 22.

So the calculation is:

nine full stop two multiplication nine full stop five multiplication one zero super one zero postfix multiplication one equation left hand side zero super 12 equals right hand side 87 full stop four multiplication one zero super 22 km
equation left hand side prefix of equals right hand side eight full stop 74 multiplication one zero super one postfix multiplication one zero super 22 km
prefix of equals eight full stop 74 multiplication one zero super open one plus 22 close km
equation left hand side prefix of equals right hand side eight full stop 74 multiplication one zero super 23 km

That means the width of the universe is 8.74 prefix multiplication of 10 super 23 km.

We have a rule for multiplying two numbers with the same base number, but what about division?

Consider 10 super six division 10 squared. This can be written as a fraction, and then canceled, as follows:

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

Therefore, equation left hand side 10 super six division 10 squared equals right hand side 10 super four.

This time you can get the same result by subtracting the powers: equation sequence 10 super six division 10 squared equals 10 super open six minus two close equals 10 super four

Try working out three super eight division three cubed the “long way,” and then by subtracting the powers. Are you convinced that the following rule works?

To divide two numbers with the same base number, subtract the powers.

Make a note of these rules about powers with the same base in your math notebook - remembering to highlight that they apply only with the same base number.

Activity Symbol Activity: Dividing Numbers with the Same Base Number

Without using your calculator, work out each of the following.

(a) four super seven division four super five

Solution Symbol

Answer

(a) equation sequence four super seven division four super five equals four super open seven minus five close equals four squared equals four multiplication four equals 16

(b) three super four division three super seven

Solution Symbol

Answer

(b) equation sequence three super four division three super seven equals three super open four minus seven close equals three super negative three equals one divided by three cubed equals one divided by three multiplication three multiplication three equals one divided by 27

(c) two super four division two super two minus

Solution Symbol

Answer

(c) equation sequence two super four division two super two minus equals two super open four minus open two minus close close equals two super four plus two equals two super six equals two multiplication two multiplication two multiplication two multiplication two multiplication two equals 64

(d) 10 super two minus division 10 squared

Solution Symbol

Answer

(d) equation sequence 10 super two minus division 10 squared equals 10 super open two minus minus two close equals 10 super four minus equals one divided by 10 multiplication 10 multiplication 10 multiplication 10 equals one divided by 10 comma 000 equals 0.0001

10.1.2 Scientific Notation

10.1.4 Zero as a Power