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

[FREE] Anything to the Power of 0 Worksheet (Grade 8)

[FREE] Anything to the Power of 0 Worksheet (Grade 8)

[FREE] Anything to the Power of 0 Worksheet (Grade 8)

Use this worksheet to check your grade 8 students’ understanding of raising terms to the power of 0. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE
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[FREE] Anything to the Power of 0 Worksheet (Grade 8)

[FREE] Anything to the Power of 0 Worksheet (Grade 8)

[FREE] Anything to the Power of 0 Worksheet (Grade 8)

Use this worksheet to check your grade 8 students’ understanding of raising terms to the power of 0. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

Patterns in 1 × the base

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

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.

What is raising anything to the power of 0?

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}.

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.

  • 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}.

Practice anything to the power of 0 questions

1. Simplify x^{0}.

0
GCSE Quiz False

x
GCSE Quiz False

1
GCSE Quiz True

x^{0}
GCSE Quiz False

x^{0}=1 always.

2. Simplify 8x^{0}.

8
GCSE Quiz True

80
GCSE Quiz False

1
GCSE Quiz False

8x
GCSE Quiz False

x^{0}=1 always.

 

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

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

1
GCSE Quiz False

4x
GCSE Quiz False

4
GCSE Quiz True

x^{4}
GCSE Quiz False
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
GCSE Quiz False

11x
GCSE Quiz False

s^{11}
GCSE Quiz False

11
GCSE Quiz True
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
GCSE Quiz False

25
GCSE Quiz True

m^{24}
GCSE Quiz False

625
GCSE Quiz False
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
GCSE Quiz False

– \, 160
GCSE Quiz False

2.5
GCSE Quiz True

5r^0
GCSE Quiz False
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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