Anything to the power of 0

Here you will learn about raising anything to the power of 0, including an explanation and examples of how to use it when solving.

Students will first learn about raising anything to the power of 0 as part of expressions and equations in 8 th grade.

What is raising anything to the power of 0?

Raising anything to the power of \bf{0} (zeroth power) makes it equal to 1.

Let’s look at this in three different ways:

Quotient of a number divided by itself

Remember, any number divided by itself is 1.

For example,

So, x^{2}\div{x^2}=1

Using the rules of exponents, when you divide two terms with the same base you subtract the powers.

So this means that,

x^{0}=1

Patterns in 1 × the base

Another way to think about this is by looking at patterns in the expanded equations.

For example,

Which is exactly the same as,

You can think of 2^3 as telling us to do 1 multiplied by 2 however many times the exponent tells us, in this case 3 times.

If you continue this pattern, you get the following:

In other words, think of 2^0 as multiplying 1 by 2 however many times the exponent tells us, in this case zero times.

So, 2^{0}=1

Comparing other exponents of the same base

You could also think about it like this:

Each time the exponent decreases by 1, you divide the value by whatever the base is, in this case 2.

So, 2^{0}=1

Whichever way you look at it, if a term has a zero exponent (it is raised to the power of zero), its value is 1. This is the zero exponent rule or the zero property of exponents.

What is raising anything to the power of 0?

Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Expressions and Equations (8.EE.A.1)
  Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^{2}× 3^{-5}=3^{-3}=\cfrac{1}{3^{3}}=\cfrac{1}{27}.

How to solve anything to the power of 0

In order to solve anything to the power of 0\text{:}

1. Substitute any term with the power of \bf{0} as \bf{1}.
2. Solve any remaining parts of the expression.

Anything to the power of 0 examples

Example 1: no coefficient in front of base

Simplify a^{0}.

1. Substitute any term with the power of \bf{0} as \bf{1}.

2Solve any remaining parts of the expression.

There is no other part to solve.

Example 2: coefficient in front of the base

Simplify 6a^{0}.

Example 3: coefficient in front of base and constant

Simplify 50h^0+8.

Example 4: coefficient in front of base and positive powers

Simplify \cfrac{2x^{0}}{4^{2}}.

Example 5: power of 0 in a linear expression in the denominator

Simplify \cfrac{6^3}{0.5{m^0}+17.5}.

Example 6: negative exponents and decimals

Simplify 2.5x^{0}\div{2^{-2}}.

Teaching tips for anything to the power of 0

• Review any properties of exponents used in the explanations of the power of 0.

• Start with simpler examples and work your way up to more complex ones only when students are ready.

Easy mistakes to make

• Thinking a number raised to the power of \bf{0} is equal to \bf{0}
  This is a very common misunderstanding, but can be overcome by explaining that x^{n}\div{x^n}=x^{n-n}=x^{0} and x^{n}\div{x^n}=1 .

• Not realizing all real number to the power of 0 are equal to \bf{1}
  Regardless of whether it is a whole number or a decimal or a fraction, or a positive or a negative number, or a rational number (example, 4,~0.25,~\cfrac{1}{2} etc.), or an irrational number (example, \pi,~\sqrt{5},~e (Euler’s number) etc.) raising a base number or a base variable to the power 0 will give a value of 1.
  
  Raising algebraic polynomials to the power of 0 is also 1. Any exponent (sometimes called an index) that is a non-zero number will not give 1.

• Confusing integer and fractional powers
  Raising a term to the power of 2 means you square it.
  For example, a^{2}=a\times{a}.
  
  Raising a term to the power of \cfrac{1}{2} means you find the square root of it.
  For example, a^{\frac{1}{2}}=\pm \sqrt{a}.
  
  Raising a term to the power of 3 means you cube it.
  For example, a^{3}=a \times a \times a.
  
  Raising a term to the power of \cfrac{1}{3} means you find the cube root of it.
  For example, a^{\frac{1}{3}}=\sqrt[3]{a}.

Related law of exponents lessons

• Exponential notation
• Multiplying exponents
• Dividing exponents
• Distributing exponents
• Square root
• Cube root

The next lessons are

• Scientific notation
• Quadratic graphs
• Factoring

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs.