The exponent the a number claims how plenty of times to use the number in a multiplication.

You are watching: 5 to the power of -4

In this example: 82 = 8 × 8 = 64


Fractional Exponents

But what if the exponent is a fraction?

An exponent the 12 is a square root

An exponent the 13 is a cube root

An exponent of 14 is a 4th root

And therefore on!

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Why?

Let"s watch why in an example.

First, the regulations of exponents tell us exactly how to manage exponents as soon as we multiply:


Example: What is 9½ × 9½ ?

9½ × 9½ = 9(½+½) = 9(1) = 9

So 9½ times itself provides 9.

Now, what perform we call a number that, when multiplied by itself, gives an additional number? The square root of that various other number!

See:

√9 × √9 = 9

And:

9½ × 9½ = 9

So 9½ is the exact same as √9


Example:

16¼ × 16¼ × 16¼ × 16¼ = 16(¼+¼+¼+¼) = 16(1) = 16

So 16¼ used 4 times in a multiplication provides 16,

and so 16¼ is a fourth root that 16


General Rule

It operated for ½, it operated with ¼, in fact it works generally:

x1/n = The n-th root of x

In various other words:


What around More complicated Fractions?

What about a fractional exponent choose 43/2 ?

That is yes, really saying to do a cube (3) and a square root (1/2), in any kind of order.

Let me explain.

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A fraction (like m/n) deserve to be damaged into 2 parts:

a whole number component (m) , and also a portion (1/n) component

So, due to the fact that m/n = m × (1/n) we deserve to do this:

xm/n =x(m × 1/n) =(xm)1/n = n√xm

The order does not matter, so it additionally works for m/n = (1/n) × m:

xm/n =x(1/n × m) =(x1/n)m = (n√x )m

And we get this:


Example: What is 43/2 ?

43/2 = 43×(1/2) = √(43) = √(4×4×4) = √(64) = 8

or

43/2 = 4(1/2)×3 = (√4)3 = (2)3 = 8

Either means gets the exact same result.


Example: What is 274/3 ?

274/3 = 274×(1/3) = 3√274 = 3√531441 = 81

or

274/3 = 27(1/3)×4 = (3√27 )4 = (3)4 = 81

It was definitely easier the 2nd way!


Now ... Play through The Graph!

See how smoothly the curve alters when you play with the fractions in this animation, this mirrors you the this idea the fractional exponents fits together nicely:


Things to try:

start with m=1 and also n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4 Then shot m=2 and slide n up and down to view fractions choose 2/3 and so on Now try to make the exponent −1 Lastly try increasing m, climate reducing n, climate reducing m, then increasing n: the curve should go around and also around