Introduction
10 is a unique number to multiply or divide by, because we can find the answer quickly by "jumping the decimal".
We can raise 10 to a certain power to make our problems more difficult.
When we raise 10 to a power such as 103 this means we will multiply 10 by itself three times.
103 = 10 × 10 × 10
103 = 1000
When we multiply by an exponent of 10, we need to shift our decimal point to the right as many times as the exponent of 10.
102, 103, 104,...
When we divide by an exponent of 10, we need to shift our decimal point to the left as many times as the exponent of 10.
124 ÷ 102 = 1.24
Our decimal point will always be after the ones unit place.
Multiplying by Exponents of Ten
As we saw previously when we multiply by an exponent of 10, we will shift our decimal space to the right as many times as the exponent.
No matter how large our exponent gets, we can use this method to solve any similar multiplication problem.
Rules for Multiplying by Exponents of Ten
Let's take a look at multiplying a decimal number by 10. We just need to shift the decimal one space to the right in this case.
825.3 × 10 = 8253
What happens if we multiply the same decimal number by 103?
We shifted the decimal to the right 3 times since our exponent is 3. Once we get to the last digit of our number we continue shifting the decimal point to the right and adding zeros as many times as we need to.
825.3 × 103 = 825300
Alternatively, we can think of 825.3 × 103 as 825.3 × 1000.
825.3 × 1000 = 825300
Even though the number 10 does not appear to have an exponent, there is an invisible exponent of 1 above it.
10 = 101
Dividing by Exponents of Ten
As we saw previously when we divide by an exponent of 10, we will shift our decimal space to the left as many times as the exponent.
No matter how large our exponent gets, we can use this method to solve any similar division problem.
Rules for Dividing by Exponents of Ten
Let's take a look at dividing a decimal number by 10. We just need to shift the decimal one space to the left in this case.
825.3 ÷ 10 = 82.53
What happens if we divide the same decimal number by 103?
We shifted the decimal to the left 3 times since our exponent is 3. Since we no longer have any digits before the decimal, we can add a zero at the very end.
825.3 ÷ 103 = .8253
.8253 = 0.8253
What happens if we divide the same decimal number by 104?
We shifted the decimal to the left 4 times since our exponent is 4. Since we no longer have any digits we will add a zero as we shift to the left.
825.3 ÷ 104 = .08253
.08253 = 0.08253

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