Math resources Algebra Laws of exponents

Anything to the power of 0

# 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,

5\div5=1

\cfrac{5}{6}\div\cfrac{5}{6}=1

2x\div{2x}=1

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.

x^{2}\div{x^2}=x^{2-2}=x^{0}

So this means that,

x^{0}=1

### Patterns in 1× the base

For example,

2^{3}=2\times2\times2

Which is exactly the same as,

2^{3}=1\times2\times2\times2

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:

\begin{aligned}2^{3}&=1\times2\times2\times2 \\\\ 2^{2}&=1\times2\times2 \\\\ 2^{1}&=1\times2 \\\\ 2^{0}&=1 \end{aligned}

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:

\begin{aligned}2^{2}&=1\times2\times2=4 \\\\ 2^{1}&=1\times2=2 \\\\ 2^{0}&=1 \\\\ 2^{-1}&=1\times\cfrac{1}{2}=\cfrac{1}{2} \\\\ 2^{-2}&=1\times\cfrac{1}{2}\times\cfrac{1}{2}=\cfrac{1}{4} \end{aligned}

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.

## 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}.

a^0=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}.

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

Solve any remaining parts of the expression.

### Example 3: coefficient in front of base and constant

Simplify 50h^0+8.

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

Solve any remaining parts of the expression.

### Example 4: coefficient in front of base and positive powers

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

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

Solve any remaining parts of the expression.

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

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

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

Solve any remaining parts of the expression.

### Example 6: negative exponents and decimals

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

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

Solve any remaining parts of the expression.

### Teaching tips for anything to the power of 0

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

### 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}.

### Practice anything to the power of 0 questions

1. Simplify x^{0}.

0

x

1

x^{0}

x^{0}=1 always.

2. Simplify 8x^{0}.

8

80

1

8x

x^{0}=1 always.

8\times{x^0}=8\times{1}=8

3. Simplify 2^{2}x^{0}.

1

4x

4

x^{4}
2^2{x^0}=2^2\times{1}

This is because the variable, x, raised to the power zero equals 1.

2^2\times{1}=4\times{1}=4

4. Simplify \cfrac{88s^0}{2^3}.

1

11x

s^{11}

11
88s^0=88\times{1}=88

2^{3}=2\times{2}\times{2}=8

\cfrac{88}{8}=11

5. Simplify \cfrac{5^4}{m^{0}+24}.

1

25

m^{24}

625
m^{0}+24=1+24=25

5^{4}=5\times{5}\times{5}\times{5}=625

\cfrac{625}{25}=25

6. Simplify 2^{-3}r^{5}\div(0.05r^{5}).

1

– \, 160

2.5

5r^0
2^{-3}=\cfrac{1}{2^3}=\cfrac{1}{8}

0.05r^{5}=0.05\times{r^5}\div{r^5}=r^{5-5}=r^{0}=1

2^{-3}r^{5}\div(0.05r^{5})=\left(\cfrac{1}{8}\div{0.05}\right)\times{1}=\cfrac{1}{8}\div\cfrac{1}{20}=\cfrac{1}{8}\times{20}=2.5

## Anything to the power of 0 FAQs

What does it mean to “raise a number to a power”?

This is one way to talk about exponents. It means that the number is being multiplied by itself a certain number of times. For example, 2^n is 2 multiplied by itself n times.

Is the zeroth power a positive or negative power?

Since 0 is neither positive nor negative, the same is true for the exponent 0. It is neither a positive exponent nor a negative exponent.

What is the binomial theorem?

It was developed by mathematicians to easily expand (a+b)^n expressions.

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