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### Course: Arithmetic>Unit 17

Lesson 5: Intro to exponents

# Exponents of decimals

Exponents of decimals can be calculated by multiplying the decimal number by itself as many times as the exponent indicates. When multiplying decimals, count the total number of digits to the right of the decimal points in both numbers and place the same number of digits to the right of the decimal point in the product.

## Want to join the conversation?

• how do you do 10 to the 12 power
• 10 to the 12th power would be 10 times 10 times 10 times 10 times 10 times 10 times 10 times 10 times 10 times 10 times 10 times 10. The answer would be 1000000000000
• At the end of the video sals says another way to think about it is "9 tenths of 9 tenths is 81 hundreths"

But isn't 0.9/0.9 = 1?
• "Of" refers to multiplication; He's saying 0.9*0.9 = 0.81
• 0.9 x 0.9= 0.81? I see the math and understand how it is done. But 0.81 is smaller than 0.90? How does a bigger number (0.9) multiplied by itself ended up being smaller?
• I think it's because they are decimals, so when they are multiplied, they go backwards instead of forwards. Is that helpful at all? It probably isn't. I hope I didn't confuse you more!
• A lot of the time when I'm stuck on an equation I tend to watch the video a few times and sometimes it still doesn't help because the person on the video does a lot of the math in their head and that's where it loses me is there any way for me to get one-on-one help?
• Hi Christine,
Yes, I have that problem sometimes too. Sal (in the voiceover) usually does the mental math faster than we can catch up. Unfortunately there is no feature here that you can get one-on-one help. (or not that I know of)

What I would suggest it to pause the video and try to do the math yourself on a piece of paper or in your head. If you don't get that math, you could try searching up how to do it here on Khan Academy.
• I really hate how he doesn't fully explain why you don't add the same amount of zeros to the second example here. You moved the decimal two spots to get 0.04 after multiplying 0.2 x 0.2. But why don't you do that for 0.9?
• You need to count decimal places needed. In both cases, the numbers being multiplied have 1 decimal place. 1+1 = 2 decimal places in your answer.
2x2 = 4
Since this is one digit, you need to put a zero in front of the 4 to create the 2 decimal place: 0.04

9x9 = 81
This is already a 2 digit number. So, the decimal point goes in front of the 8: 0.81

Note: This is no different then if you multiplied 0.2 x 0.12
There are 3 decimal digits. So, the answer becomes 0.024

You may find it helpful to review the lessons on multiplying decimal numbers.
• What is 0 to the power of 0?
• 0 to the power of 0 = 1
• how do you use a exponent when using a fraction, for example (2/5) and 3 is a exponent.
• You have to convert the fraction (2/5) to a decimal (0.4) than you can use the exponent (3) so 0.4x0.4x0.4= 0.064 hope this helps
• what is 24 to the power of 5
• Sorry, but I can not just give you the answer so I will tell you how to get the answer so you can also write out 24 to the power of 5 like this "24 Times 24 times 24 times 24 times 24 =" so exponents are pretty much multiplying one number as many times as the exponent says to. I hope this helped.
• Would zero to the zeroth power be one or zero?
• i think it would be just a 0
• I physically cannot comprehend why 0.2^3 (zero point two to the third power) isn't .8 and is 0.008. Even the explanation he barely provided didn't make sense. Why is anything decreasing in value after it's been multiplied? I know its a decimal and that's the reason but I don't get why it gains zeroes out of nowhere.
• Good question. The `2` in `0.2` is the tenths place value, so this number is in tenths. So when multiplying by itself you multiply tenths by tenths and this will become hundredths, and so on.

You can think of any decimal numbers less than one, such as `0.2`, as an equivalent power-of-ten fraction: `2/10`!
Recall the property of exponent of fractions (this arises from how fraction multiplication works): `(a/b)^n = (a^n)/(b^n)`.
In steps:`0.2^1 = (2^1)/(10^1) = 2/10 = 0.20.2^2 = (2^2)/(10^2) = 4/100 = 0.040.2^3 = (2^3)/(10^3) = 8/1000 = 0.0080.2^4 = (2^4)/(10^4) = 16/10000 = 0.0016` [Notice how for this iteration, the numerator (place value) become a 2-digit number, preventing the decimal rollover]

You can see how the denominator gets multiplied by 10, which translates to moving 1 decimal place to the left, or adding a 0. Hopefully now you can see why this is the case.

I hope this helps; I have no idea if it will.