Zero power with with integer and variable bases

As we learnt in Dividing Powers, when we divide index terms with like bases, we subtract the powers.

So what happens when we subtract and we are left with a power of 0? For example, 

$4^1\div4^1$41÷​41	$=$=	$4^{1-1}$41−1
	$=$=	$4^0$40

Let's write this division problem as a fraction:

$\frac{4^1}{4^1}=1$4141​=1

You'll notice that the numerator and denominator are the same so the fraction simplifies to $1$1. Notice that this will also be the case with $\frac{4^2}{4^2}$4242​ or any expression where we are dividing like bases whose indices (or powers) are the same. 

So the result we arrive at by using index laws is $4^0$40, and the result we arrive at by simplifying fractions is $1$1This must mean that: $4^0=1$40=1

If we extend this to any other base, we get the result that

$x^0=1$x0=1

for any value of $x$x.

 

Some easy rules to follow:

• Any number to the power of 0 is equal to 1

$x^0=1$x0=1

• When an expression is enclosed in brackets, then the index (or power) is applied to everything inside the brackets

 

Example

$\left(x+1\right)^0=1$(x+1)0=1

Similarly: $\left(5x\right)^0=1$(5x)0=1

If there are no brackets, then the index (or power) only applies to the number it is connected to:

$5\times x^0$5×x0	$=$=	$5\times1$5×1
	$=$=	$5$5

 

More examples

Question 1

Simplify $705^0$7050

$705^0=1$7050=1

 

Question 2

Question 3