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Negative numbers Multiplication Division Fraction to decimal Exponent rulesHere 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.

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

Let’s look at this in three different ways:

Remember, any number divided by itself is 1.

For example,

5\div5=1 \cfrac{5}{6}\div\cfrac{5}{6}=1 2x\div{2x}=1So, 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

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

For example,

2^{3}=2\times2\times2Which is exactly the same as,

2^{3}=1\times2\times2\times2You 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

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.

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}.

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

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

Simplify a^{0}.

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

2**Solve any remaining parts of the expression.**

There is no other part to solve.

Simplify 6a^{0}.

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

6a^0=6\times{1} because a^0=1.

**Solve any remaining parts of the expression.**

6\times{1}=6

6a^0=6, no matter the value of a.

Simplify 50h^0+8.

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

50h^0=50\times{1} because h^0=1.

**Solve any remaining parts of the expression.**

(50\times{1})+8=58

50h^0+8=58, no matter the value of h.

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

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

2x^0=2\times{1} because x^0=1.

**Solve any remaining parts of the expression.**

\cfrac{2\times{1}}{4^2}=\cfrac{2}{16}=\cfrac{1}{8}

\cfrac{2x^0}{4^2}=\cfrac{1}{8}, no matter the value of x.

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

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

0.5m^0=0.5\times{1} because m^0=1.

**Solve any remaining parts of the expression.**

\cfrac{6^3}{0.5\times{1}+17.5}=\cfrac{216}{18}=12

\cfrac{6^3}{0.5m^{0}+17.5}=12, no matter the value of m.

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

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

2.5x^0=2.5\times{1} because x^0=1.

**Solve any remaining parts of the expression.**

\begin{aligned}&(2.5\times{1})\div{2^{-2}} \\\\ &=2.5\div{2^{-2}} \\\\ &=2.5\div\cfrac{1}{2^2} \\\\ &=2.5\div\cfrac{1}{4} \\\\ &=2.5\times{4} \\\\ &=10 \end{aligned}

2.5x^0\div{2^{-2}}=10, no matter the value of x.

**Step-by-step guide:** Negative exponents

- 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.

**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}.

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

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

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.

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.

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

- Scientific notation
- Quadratic graphs
- Factoring

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