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Standard 6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents

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Here's a funny video to demonstrate one reason you might want to know these things.

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When not knowing Math can cost you $15,000

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Check out this website for an introduction into exponents.

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The exponent of a number says how many times to use the number in a multiplication.

In words: 8 ² could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"
Exponents are also called Powers or Indices.
Some more examples:

Example: 5³ = 5 × 5 × 5 = 125

In words: 5³ could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

Example: 2⁴ = 2 × 2 × 2 × 2 = 16

In words: 2⁴ could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

Exponents make it easier to write and use many multiplications

Example: is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9

You can multiply any number by itself as many times as you want using exponents.

Try here:

algebra/images/exponent-calc.js

So in general:

a n tells you to multiply a by itself,
so there are n of those a's:

Another Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.

Example: 2^4 is the same as 2⁴

Negative Exponents

Negative? What could be the opposite of multiplying? Dividing!

So we divide by the number each time, which is the same as multiplying by

We can continue on like this:

Example: 5⁻³ = 1 1 1 5 = 0.0085 × 5 ×

But it is often easier to do it this way:

5⁻³ could also be calculated like:

Negative? Flip the Positive!

That last example showed an easier way to handle negative exponents:

Calculate the positive exponent (an)

Then take the 1/aReciprocal (i.e. n)

More Examples:

What if the Exponent is 1, or 0?

It All Makes Sense

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern:

Example: Powers of 5

.. etc..

5²
5 × 5

25

5¹
5

5

5⁰
1

1

5⁻¹
1 5

0.2

5⁻²
1 1 5 5 ×

0.04

.. etc..

Be Careful About Grouping

To avoid confusion, use parentheses () in cases like this:

305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864

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The exponent of a number says how many times to use the number in a multiplication. In words: 8 2 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared" Exponents are also called Powers or Indices.

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Standard: 6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.

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Lets do an example problem.

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10³=?

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Lets write it out in expanded form (see below)

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If we multiply 10 x 10 x 10 = ? we get 1,000! You can check it with calculator below.

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Calculator Use

This is an online calculator for exponents. Calculate the power of large base integers and real numbers. You can also calculate numbers to the power of large exponents less than 2000, negative exponents, and real numbers or decimals for exponents.

For larger exponents try the Large Exponents Calculator

For instructional purposes the solution is expanded when the base x and exponent n are small enough to fit on the screen. Generally, this feature is available when base x is a positive or negative single digit integer raised to the power of a positive or negative single digit integer. Also, when base x is a positive or negative two digit integer raised to the power of a positive or negative single digit integer less than 7 and greater than -7.

For example, 3 to the power of 4:

\( x^n = \; 3^{4} \)

\( = \;3 \cdot 3 \cdot 3 \cdot 3 \)

\( = 81 \)

For example, 3 to the power of -4:

\( x^n = \;3^{-4} \)

\( = \dfrac{1}{3^{4}} \)

\( = \; \dfrac{1}{3 \cdot 3 \cdot 3 \cdot 3} \)

\( = \; \dfrac{1}{81} \)

\( = 0.012346 \)

Exponent Notation:

Note that -4² and (-4)² result in different answers: -4² = -1 * 4 * 4 = -16, while (-4)² = (-4) * (-4) = 16. If you enter a negative value for x, such as -4, this calculator assumes (-4)n.

"When a minus sign occurs with exponential notation, a certain caution is in order. For example, (-4) ² means that -4 is to be raised to the second power. Hence (-4) ² = (-4) * (-4) = 16. On the other hand, -4 ² represents the additive inverse of 4 ². Thus -4 ² = -16. It may help to think of -x ² as -1 * x ² ..."[1]

Examples:

3 raised to the power of 4 is written 3⁴ = 81.

-4 raised to the power of 2 is written (-4)² = 16.

-3 raised to the power of 3 is written (-3)³ = -27. Note that in this case the answer is the same for both -3³ and (-3)³ however they are still calculated differently. -3³ = -1 * 3 * 3 * 3 = (-3)³ = -3 * -3 * -3 = -27.

For 0 raised to the 0 power the answer is 1 however this is considered a definition and not an actual calculation.

Exponent Rules:

\( x^m \cdot x^n = x^{m+n} \)

\( \dfrac{x^m}{x^n} = x^{m-n} \)

\( (x^m)^n = x^{m \cdot n} \)

\( (x \cdot y)^m = x^m \cdot y^m \)

\( \left(\dfrac{x}{y}\right)^m = \dfrac{x^m}{y^m} \)

\( x^{-m} = \dfrac{1}{x^m} \)

\( \left(\dfrac{x}{y}\right)^{-m} = \dfrac{y^m}{x^m} \)

\( 0^0 = 1 \; (definition) \)

\( if \; x^m = y \; then \; y = \sqrt[m]{x} = y^{\frac{1}{m}} \)

\( x^{\frac{m}{n}} = \sqrt[n]{x^m} \)

References

[1] Algebra and Trigonometry: A Functions Approach; M. L. Keedy and Marvin L. Bittinger; Addison Wesley Publishing Company; 1982, page 11.

For more detail on Exponent Theory see Exponent Laws.

To calculate fractional exponents use our Fractional Exponents Calculator.

To calculate root or radicals use our Roots Calculator.

Last updated: August 17, 2023

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Calculator for Exponents. Find the power of large integers and decimals. Negative exponents, decimal exponents and large exponents. Exponent Rules. Shows solution for small integers to the power of small integers.

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12⁰=?
14.6795⁰=?
567034⁰=?
(Hint, they should all be the same answer)

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Why any number to the zero power always gives a one?

Question Date: 2011-09-01

Answer 1:

This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about what a power means. When we raise a number to the nth power, that really means that we multiply that number by itself n times, so for example, 2 ² = 2*2 = 4, 2 ³ = 2*2*2 = 8, 3 ⁴ = 3*3*3*3 = 81, and so on. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all! What does that mean? Well, let's go even farther back to the simplest case: addition. What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same. In short, 0 is the only number such that for any number x, x + 0 = x. So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number. In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.

Answer 2:

It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always (well almost always) good reason for them. Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question:

Mathematics was initially developed to describe relationships between everyday quantities (generally whole numbers) so the best way to think about powers like a b ('a' raised to the 'b' power) is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'. For example, 2 ³ is 8. Why? There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2:(1,1,1) (1,1,2) (1,2,1) (2,1,1) (2,1,2) (2,2,2) (1,2,2) (2,2,1).

So what does 3⁰ represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table? Just one... don't put anything on the table and that's your only option. Therefore it's consistent to say 3 ⁰ = 1.

There are other reasons why a ⁰ has to be 1 - for example, you may have heard the power rule: a(b+c) = a b * a c. What happens if b = 0? Well I know that a(0+c) is the same as a c and by the first formula this is also a ⁰ * a c. What choice for a ⁰ make sense to satisfy a c = a ⁰ * a c? Clearly a ⁰ = 1. I know this sounds a little fishy since we started with a rule I could have just made up (which is why I gave the other reason first), but these formulas are all consistent and there is never any magic step, I promise! You can always work backwards to prove a(b+c) = a b * a c etc etc.
Hope this helps!

Answer 3:

Any number to the zero power always gives one.

One rule for exponents is that exponents add when you have the same base. So if you have a number, x, and exponents, a and b, then:
x a * x b = x(a+b)
So then if we make one of the exponents negative:x a * x-b = x(a-b)
And if the exponents are the same magnitude (a = b)x a * x-b = x a * x-a = x(a-a) = x ⁰

Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:
x -a = 1/x a
So, we can also write x a * x-a in a different way:
x a * x-a = x a * 1/x a = x a/x a
And a number divided by itself is always 1 so:
x a * x-a = x a* 1/x a = x a/x a = 1:
So now we've shown that:
x a * x-a = x(a-a) = x ⁰
and
x a * x-a = x a * 1/x a:
This means that any number x⁰ = 1.
If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:

If we plug in numbers, (for example let x = 5, a = 2, and b = 4) then:
One rule for exponents is that exponents add when you have the same base.
5 ² * 5 ⁴ = 5⁽²⁺⁴⁾ = 5 ⁶ = 15625
So then, if we make one of the exponents negative:
5 ² * 5⁻⁴ = 5⁽²⁻⁴⁾ = 5⁻² = 0:04
And if the exponents are the same magnitude:
5 ² * 5⁻² = 5⁽²⁻²⁾ = 5 ⁰

Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:
5⁻² = 1/5 ² = 0:04
So we can also write 5 ² * 5⁻² in a different way:
5 ² * 5⁻² = 5 ² * 1/5 ² = 5 ²/5 ² = 25/25

And a number divided by itself is always 1 so:
5 ² * 5⁻² = 5 ² * 1/5 ² = 5 ²/5 ² = 25/25 = 1

So now we've shown that:
5 ²*5⁻² = 5⁽²⁻²⁾ = 5 ⁰
and
5 ² * 5⁻² = 5 ²/5 ² = 1
This means that 5 ⁰ = 1.

This works for any number x that you want to plug in except for x = 0,because 0/0 is indeterminate (it is like dividing zero by zero).

Answer 4:

Let's look at what it means to raise a number to a certain power: it means to multiply that number by itself a certain number of times. Three to the second power is three multiplied by itself 2 times, or 3*3=9. Let's look at a few examples:

3 ⁵ = 3*3*3*3*3 = 243
3 ⁴ = 3*3*3*3 = 81
3 ³ = 3*3*3 = 27
3 ² = 3*3 = 9
3 ¹ = 3 = 3

But how do you go from 3 ¹ to 3 ⁰? If you look at the pattern, you can see that each time we reduce the power by 1 we divide the value by 3. Using this pattern we can not only find the value of 3 ⁰, we can find the value of 3 raised to a negative power! Here are some examples:

3 ⁰ = 3/3 = 1
3⁻¹ = 1/3 = 0.3333... (this decimal repeats forever)
3⁽⁻²⁾ = 1/3/3 = 0.1111...
3⁽⁻³⁾ = 1/3/3/3 = 0.037037...

No matter what number we use when it is raised to the zero power it will always be 1. Suppose instead of 3 we used some number N, where N could even be a decimal. N ¹=N, and to reduce the power by 1 we divide by N, soN ⁰=N ¹/N = N ²/2N =1.

Notice that 3⁽⁻¹⁾ is the same as 1/(3 ¹), 3⁽⁻²⁾ is the same as 1/(3⁽²⁾),and so on. This gives us a useful property of exponents, namely that a(-b) is the same as 1/(a b).

Answer 5:

Heres a quick demonstration of why any number (except zero) raised to the zero power must equal 1. As an example we will let that any number be the number 3.

Note that:
3 ¹ = 3 = 3
3 ² = 3*3 = 9
3 ³ = 3*3*3 = 27
3 ⁴ = 3*3*3*3 = 81
And so on

Youll notice that 3 ³=(3 ⁴)/3, 3 ²=(3 ³)/3, 3 ¹=(3 ²)/3
In other words, 3(n-1)=(3 n)/3
So 3 ⁰=(3 ¹)/3=3/3=1

This same reasoning will work for any number (not just 3), except the number 0. It wont work for 0 because you cant divide by 0. Lets call any number x:

x(n-1)=x n/x
So x ⁰ = x⁽¹⁻¹⁾ = x ¹/x = x/x = 1

Answer 6:

of the answer is that this is how we've defined powers to be.

Raising something to a power greater than zero means multiplying it by itself a number of times equal to the power. So, for instance,

2 ¹ = 2
2 ² = 2 x 2 = 4
2 ³ = 2 x 2 x 2 = 8
and so on.

Now, you can multiply anything by 1 and it will still be the same thing, and likewise you can divide anything by 1 and it will still be the same. Therefore:
2 ¹ = 2 x 1 = 2
2 ² = 2 x 2 x 1 = 4
2 ³ = 2 x 2 x 2 x 1 = 8
You see I've just multiplied everything by 1.

Now, also note that if you raise something to a negative power, then you take the reciprocal of that something:

2⁻¹ = 1/2
2⁻² = 1 /(2x2) = 1/4
2⁻³ = 1/8

And so on. Again, we can multiply by everything by 1:
2⁻¹ = 1 x 1/2

Now, what happens when the power is zero?

Well, you're not multiplying by anything, except the 1 you started with. You're not dividing by anything, except the 1 you started with. So, what you're left over with is 1.

Now, here is the slightly more mathematically sophisticated version: when you raise something to a power, what you do is take 1 and multiply it by the base of the power a number of times equal to the power. So, by definition, raising something to the power of zero means you start with 1, and then don't multiply it by anything. So, naturally, 1 is what you're left over with.

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Why any number to the zero power always gives a one? Question Date: 2011-09-01 Answer 1: This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about what a power means.