Power Of 0 = 1 (Explain Please)

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In summary, any number with a power of 0 has a value of 1 because when you divide a number by itself, the result is always 1. This is due to the rule that when multiplying exponents with the same base, you add the exponents. When the exponent is 0, it becomes a number divided by itself, which equals 1. This also applies when subtracting exponents with the same base, as a number divided by itself is also 1. However, it should be noted that this rule does not apply to 0^0, as it is undefined.
  • #1
Nick-30
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Homework Statement


For part of my homework i need to explain:
Why does any number with a power of zero have a value of 1?


Homework Equations


100^0=1
81^0=1
5^0=1


The Attempt at a Solution

 
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  • #2
Look at the exponential sequence if you write down the positive and negative powers of a number
eg 3:
3^-3 = 1/27
3^-2 = 1/9
3^-1 = 1/3
3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27

What's the pattern between each value?

edit fixed formating
 
Last edited:
  • #3
I really don't know. Could you please give me the answer.
 
  • #4
Check what happens if you multiply 3*1/27, 3*1/9...
 
  • #5
Yeah but how do i explain why does any number with a power of 0 have a value of 1
 
  • #6
1. Do you know how to expand [itex]n^{b+c}[/itex]?

2. Now, if you set [itex]b+c=0[/itex], write b in terms of c, and repeat step 1, what do you get?
 
  • #7
Nick-30 said:
Yeah but how do i explain why does any number with a power of 0 have a value of 1

Because if you divide "any number" by itself you get 1 and if you multiply "1/any number" by "any number". Just look at the above example with any other number than 3.
 
  • #8
You know the rules for adding/subtracting exponents?

(a^b)*(a^c) = a^(b+c)

and

(a^b)/(a^c) = a^(b-c).

What happens when b = c?

You get (a^b)/(a^b) = a^(b-b) = a^0. But if you look to the LHS, you will see we have a number divided by itself which is 1.

Incidentally, it is not true that 'any number' to the power of 0 = 1: 0^0 is undefined.
 
  • #9
As others explained, ab*ac=ab+c. Now try for example:
3-1*31=3/3=1
Same if you write as:
3-1+1=30=1.

Do you know how to explain now?
 

1. What is the concept of "Power Of 0 = 1"?

The concept of "Power Of 0 = 1" refers to the mathematical rule that any number raised to the power of 0 is equal to 1. This means that 0 raised to any power, including 0, results in 1 as the answer.

2. Why does anything raised to the power of 0 equal 1?

This is a fundamental rule in mathematics that is based on the concept of exponents. When a number is raised to a certain power, it is essentially being multiplied by itself that number of times. However, when a number is raised to the power of 0, it means that it is being multiplied by itself 0 times. Anything multiplied by 0 is equal to 0, so the answer is always 1.

3. How does the concept of "Power Of 0 = 1" apply to real-life situations?

The concept of "Power Of 0 = 1" may not have direct applications in real-life situations, but it is a fundamental rule in mathematics that is used in various calculations and equations. For example, it is used in calculating probabilities and in solving logarithmic equations.

4. Can the concept of "Power Of 0 = 1" be proven?

Yes, the concept of "Power Of 0 = 1" can be proven using mathematical proofs and demonstrations. It is a well-established rule in mathematics and is widely accepted by the scientific community.

5. Are there any exceptions to the rule of "Power Of 0 = 1"?

No, there are no exceptions to this rule. It holds true for all real numbers and is a fundamental concept in mathematics.

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