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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(15 votes)
- 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(44 votes)

- 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?(9 votes)- "Of" refers to multiplication; He's saying 0.9*0.9 = 0.81(27 votes)

- 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?(12 votes)
- 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!(17 votes)

- 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?(7 votes)
- 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.(18 votes)

- 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?(9 votes)
- 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.(12 votes)

- how do you use a exponent when using a fraction, for example (2/5) and 3 is a exponent.(5 votes)
- 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(4 votes)

- Would zero to the zeroth power be one or zero?(5 votes)
- i think it would be just a 0(2 votes)

- What is 0 to the power of 0?(7 votes)
- Certainly! The confusion you're experiencing is quite common, and it all revolves around the concept of exponents in mathematics. Let's break it down to clarify why \(0^0 = 1\) but \(0^1 = 0\).

### Understanding Exponents:

1. ***General Rule for Exponents***:

- For any non-zero number \(a\) and positive integer \(n\), \(a^n\) means multiplying \(a\) by itself \(n\) times. For example, \(2^3 = 2 \times 2 \times 2 = 8\).

2. ***Exponent of Zero***:

- When we deal with \(0^n\) where \(n\) is a positive integer, the result is always 0. This is because multiplying zero by itself any number of times still results in zero. So, \(0^1 = 0\), \(0^2 = 0\), and so on.

### The Case of \(0^0\):

1. ***Indeterminate Form***:

- The expression \(0^0\) is considered an indeterminate form in mathematics. This means it doesn't fit neatly into the rules we use for other exponents and can be interpreted in different ways depending on the context.

2. ***Context and Definitions***:

- In some contexts, particularly in combinatorics and certain algebraic structures, \(0^0\) is defined to be 1. This is because it simplifies various mathematical formulas and helps maintain consistency in certain mathematical theories. For instance, the number of ways to choose 0 items from a set of 0 items is 1, which aligns with this definition.

3. **Why Different Results?**:

- For \(0^1\), the rule is straightforward: \(0\) raised to any positive exponent is always \(0\).

- For \(0^0\), it’s less about a straightforward application of rules and more about convention and context. The value of \(0^0\) is defined as 1 in many contexts to avoid complications and to make formulas work smoothly.

In summary, \(0^1 = 0\) because multiplying zero by itself once results in zero. \(0^0\), however, is a special case where the value is often defined as 1 for practical reasons, though mathematically it’s an indeterminate form and can be interpreted differently depending on the context.(1 vote)

- i dont understand how when its 0.2 it gets pushed into the 0.04. like with an added 0.

anyone care to explain?(4 votes)- Think of
**0.2 x 0.2**as**1/5th of 0.2**, because 0.2 is a fifth of a whole number. What goes into 0.2*five*times? Your answer would be**0.04**!(4 votes)

- what is 24 to the power of 5(2 votes)
- 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.(7 votes)

## Video transcript

- [Instructor] What we're
going to do in this video is get some practice evaluating
exponents of decimals. So let's say that I have
0.2 to the third power. Pause this video, see
if you can figure out what that is going to be. Well, this would just mean
that if I take something to the third power, that means
I take three of that number and I multiply them together. So it's 0.2 times 0.2 times 0.2. Well, what is this going to be equal to? Well, if I take 0.2 times
0.2, that is going to be 0.04. One way to think about
it, two times two is four and then I have one
number behind the decimal to the right of the decimal here. I have another digit to
the right of the decimal right over here, so my product
is going to have two digits to the right of the
decimal, so it'd be 0.04. And then if I were to
multiply that times 0.2, so if I were to multiply that together what is that going to be equal to? Well, four times two is equal to eight and now I have one, two, three numbers to the right of the decimal point, so my product is going to
have one, two, three numbers to the right of the decimal point. So now that we've had a little
bit of practice with that, let's do another example. So let's say that I were to
ask you, what is 0.9 squared? Pause this video and see
if you can figure that out. All right, well this is just
going to be 0.9 times 0.9. And what's that going to be equal to? Well, you could just say nine times nine is going to be equal to
81, and so, let's see, in the two numbers that I'm multiplying I have a total of one, two
numbers, or two digits, to the right of the decimal point so my answer's going
to have one, two digits to the right of the decimal point. So put the decimal right over there and I'll put the zero, so 0.81. Another way to think about it
is nine-tenths of nine-tenths is 81 hundredths, but there you go. Using exponents, or taking
exponents of decimals is the same as when we're
taking it of integers. It's just in this case you just have to do a little bit of decimal multiplication.