Why is 10 to the power of 0 equals 1?

Answer 1

Any number to the power 0 is 1 - except 00, which is undefined.

This can't really be "proven", but the definition makes sense: you may already know that, for example, 102 x 103 = 105, in other words, you can add the exponents. So, you would also have 105 x 100 = 105; the number in the middle must obviously be equal to 1.

Another way to look at it is looking at this sequence:

103 = 1000
102 = 100
101 = 10
100 = 1

The exponent in the left part is reduced by 1 each time, the number at the right, decreased by a factor of 10 each time. Thus, it makes sense to define 100 (or any other number to the power zero) as 1.

Note that this is no proof; the powers are defined so that x0 = 1 (for x not equal to zero). The above only shows that definition is reasonable.

Any number to the power 0 is 1 - except 00, which is undefined.

This can't really be "proven", but the definition makes sense: you may already know that, for example, 102 x 103 = 105, in other words, you can add the exponents. So, you would also have 105 x 100 = 105; the number in the middle must obviously be equal to 1.

Another way to look at it is looking at this sequence:

103 = 1000
102 = 100
101 = 10
100 = 1

The exponent in the left part is reduced by 1 each time, the number at the right, decreased by a factor of 10 each time. Thus, it makes sense to define 100 (or any other number to the power zero) as 1.

Note that this is no proof; the powers are defined so that x0 = 1 (for x not equal to zero). The above only shows that definition is reasonable.

Any number to the power 0 is 1 - except 00, which is undefined.

This can't really be "proven", but the definition makes sense: you may already know that, for example, 102 x 103 = 105, in other words, you can add the exponents. So, you would also have 105 x 100 = 105; the number in the middle must obviously be equal to 1.

Another way to look at it is looking at this sequence:

103 = 1000
102 = 100
101 = 10
100 = 1

The exponent in the left part is reduced by 1 each time, the number at the right, decreased by a factor of 10 each time. Thus, it makes sense to define 100 (or any other number to the power zero) as 1.

Note that this is no proof; the powers are defined so that x0 = 1 (for x not equal to zero). The above only shows that definition is reasonable.

Any number to the power 0 is 1 - except 00, which is undefined.

This can't really be "proven", but the definition makes sense: you may already know that, for example, 102 x 103 = 105, in other words, you can add the exponents. So, you would also have 105 x 100 = 105; the number in the middle must obviously be equal to 1.

Another way to look at it is looking at this sequence:

103 = 1000
102 = 100
101 = 10
100 = 1

The exponent in the left part is reduced by 1 each time, the number at the right, decreased by a factor of 10 each time. Thus, it makes sense to define 100 (or any other number to the power zero) as 1.

Note that this is no proof; the powers are defined so that x0 = 1 (for x not equal to zero). The above only shows that definition is reasonable.

Answer 2

It comes from the law of indices. One of the laws states that

xa * xb = x(a+b)

Now, if you put b = 0 in that, you get

xa * x0 = x(a+0)

but a+0 = a so the right hand side is xa

That is, xa * x0 = xa

For this to be true, x0 must be 1.

In your particular example, put x = 10.

Answer 3

Any number to the power 0 is 1 - except 00, which is undefined.

This can't really be "proven", but the definition makes sense: you may already know that, for example, 102 x 103 = 105, in other words, you can add the exponents. So, you would also have 105 x 100 = 105; the number in the middle must obviously be equal to 1.

Another way to look at it is looking at this sequence:

103 = 1000
102 = 100
101 = 10
100 = 1

The exponent in the left part is reduced by 1 each time, the number at the right, decreased by a factor of 10 each time. Thus, it makes sense to define 100 (or any other number to the power zero) as 1.

Note that this is no proof; the powers are defined so that x0 = 1 (for x not equal to zero). The above only shows that definition is reasonable.