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Zero power with variable bases


Previously we have looked at how to evaluate terms with a zero power, where the base of the term is a number. Remember that any number raised to a zero power is equal to $1$1.

As with all of the other index laws, once we are familiar with working on numbers, we can apply the same properties of the zero power to bases that are algebraic variables or even algebraic expressions.

The zero power law

For any numeric or algebraic expression $a$a, the zero power law tells us that


We can use algebraic variables in place of where we have previously used numbers. All this means is that we are replacing the specific number in the base with some letter of the alphabet.

Worked example

Simplify $g^5\div g^5$g5÷​g5 by first writing the expression in expanded form.

Think: This expression involves the division of one algebraic term by another. If we write it as a fraction we can see that the numerator and the denominator are the same.


$g^5\div g^5$g5÷​g5 $=$= $\frac{g^5}{g^5}$g5g5  
  $=$= $\frac{g\times g\times g\times g\times g}{g\times g\times g\times g\times g}$g×g×g×g×gg×g×g×g×g (We can cancel each similar factor in the numerator and denominator)
  $=$= $\frac{1}{1}$11  
  $=$= $1$1  

Reflect: Recall that $a^m\div a^n=\frac{a^m}{a^n}=a^{m-n}$am÷​an=aman=amn, so that we could have written $g^5\div g^5$g5÷​g5 as $g^{5-5}=g^0$g55=g0. Now we can see that $g^0=1$g0=1. This should be no surprise; the initial expression asks us "What do we get when we divide $g^5$g5 by itself?", to which the answer is simply $1$1, since anything divided by itself is equivalent to $1$1.

Worked example

Simplify $\left(14p\right)^0$(14p)0.

Think: The presence of brackets in the expression tells us that $14p$14p is the base and $0$0 is the power. The power acts on both the number $14$14 and the variable $p$p.

Do: The expression simplifies to $\left(14p\right)^0=1$(14p)0=1.

Reflect: Take note of where brackets occur in an expression, as they can help us determine the correct order of operations when simplifying or evaluating. For example, in the expression $14p^0$14p0 the base is $p$p and the power is $0$0. Since $p^0=1$p0=1, this would simplify to $14\times1=14$14×1=14, which is different to the value of $\left(14p\right)^0$(14p)0 that we found above.

Practice questions

Question 1

Simplify: $\left(a^0\right)^{79}$(a0)79

Question 2

Simplify $\frac{45x^7y^7}{9x^7y^2}$45x7y79x7y2.

Question 3

Evaluate: $\left(7m^0+4\right)^2$(7m0+4)2

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