10 to the power of as a fraction

Video transcript

Let's go through more exponent examples. So to warm up, let's think about taking a fraction to some power. So let's say I have 2/3, and I want to raise it to the third power here. Now, we've already learned there are two ways of thinking about this. One way is to say let's take three 2/3's. So that's one 2/3, two 2/3's, and three 2/3's. So that's one, two, three, 2/3. And then we multiply them. And we will get-- let's see, the numerator will be 2 times 2 times 2, which is 8. And the denominator's going to be 3 times 3 times 3 times 3, which is equal to 27. Now, the other way of viewing this is you start with a 1, and you multiply it by 2/3 three times. So you multiply by 2/3 once, twice, three times. You will get the exact same result here. So let's do one more example like that. So lets say I had 4/9, and I want to square it. When I raise something to the second power, people often say, you're squaring it. Also, raising something to the third power, people sometimes say, you're cubing it. But let's raise 4/9 to the second power. Let's square it. And I encourage you to pause the video and work this out yourself. Well, once again, you could view this as taking two 4/9's and multiplying them. Or you could view this as starting with a 1, and multiplying it by 4/9 two times. Either way, your numerator is going to be 4 times 4, which is 16. And your denominator is going to be 9 times 9, which is equal to 81.

Calculator Use

Use this calculator to find the fractional exponent of a number x. With fractional exponents you are solving for the dth root of the number x raised to the power n. For example, the following are the same:

\( 4^{\frac{3}{2}} = \sqrt[2]{4^{3}} \)

and since 4 cubed equals 64 we get

\( = \sqrt[2]{64} = \pm 8 \)

Notes on Fractional Exponents:

This online calculator puts calculation of both exponents and radicals into exponent form.

• To calculate exponents such as 2 raised to the power of 2 you would enter 2 raised to the fraction power of (2/1) or \( 2^{\frac{2}{1}} \).
• To calculate radicals such as the square root of 16 you would enter 16 raised to the power of (1/2).
• To calculate combined exponents and radicals such as the 4th root of 16 raised to the power of 5 you would enter 16 raised to the power of (5/4) or \( 16^{\frac{5}{4}} \) where x = 16, n = 5 and d = 4.
• If you try to take the root of a negative number your answer may be NaN = Not a Number.

For more detail on Exponent Theory see Mathworld Exponent Laws.

Explanation:

To write the number as a fraction we can use the following identity:

#a^-b=1/a^b#

#10^-4=1/10^4=1/10000#

To change the fraction whose denominator is a power of #10# to a decimal you can write the numerator and then move the decimal point as many places to the left as the number of zeros is which appear in the fraction's denominator.

#1/10000=0.0001#

The exponent (or index or power) of a number says
how many times to use the number in a multiplication.

102 means 10 × 10 = 100

(It says 10 is used 2 times in the multiplication)

Example: 103 = 10 × 10 × 10 = 1,000

• In words: 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

Example: 104 = 10 × 10 × 10 × 10 = 10,000

• In words: 104 could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4"

You can multiply any number by itself as many times as you want using this notation (see Exponents), but powers of 10 have a special use ...

Powers of 10

"Powers of 10" is a very useful way of writing down large or small numbers.

Instead of having lots of zeros, you show how many powers of 10 will make that many zeros

Example: 5,000 = 5 × 1,000 = 5 × 103

5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000

Can you see that 103 is a handy way of making 3 zeros?

Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.

Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 1030 kg.

It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg

(And very easy to make a mistake counting the zeros!)

Example: A Light Year (the distance light travels in one year)

It is easier to use 9.461 × 1015 meters, rather than 9,461,000,000,000,000 meters

It is commonly called Scientific Notation, or Standard Form.

Other Way of Writing It

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

Example: 3 × 10^4 is the same as 3 × 104

• 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Calculators often use "E" or "e" like this:

Example: 6E+5 is the same as 6 × 105

• 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 104

• 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

The Trick

While at first it may look hard, there is an easy "trick":

The index of 10 says ...

... how many places to move the decimal point to the right.

Example: What is 1.35 × 104 ?

You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500

But it is easier to think "move the decimal point 4 places to the right" like this:

Negative Powers of 10

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

A negative power means how many times to divide by the number.

Example: 5 × 10-3 = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005

Just remember for negative powers of 10:

For negative powers of 10, move the decimal point to the left.

So Negatives just go the other way.

Example: What is 7.1 × 10-3 ?

Well, it is really 7.1 x (1/10 × 1/10 × 1/10) = 7.1 × 0.001 = 0.0071

But it is easier to think "move the decimal point 3 places to the left" like this:

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

Summary

The index of 10 says how many places to move the decimal point. Positive means move it to the right, negative means to the left. Example:

How do you power to a fraction?

What is 10 to the power of 6 as a fraction?

What is 10 4 as a fraction?

What is 10 to the power of negative 4 as a decimal?