Subscribe to Homeschool Math Newsletter - filled with math teaching information

Latest from my blog
This is where you'll find the latest happenings, news, & ideas in math teaching

Math teaching videos
My videos at YouTube show you how to teach concepts.

Hover your mouse above to open a menu of various worksheets you can generate for free!

Advice, reviews, and resources to help you choose a math curriculum!

Games you can play online, interactive tutorials, fun math websites and more. Arranged by topic/level for ease of use.

Learn how to TEACH concepts or about general concerns in math education.

Reviews
In-depth reviews of math products

Math help & tutoring
A list of free message boards, math help websites, and online tutoring services.

My Amazon Store
See some math products I recommend.

I have two games on my site, plus links to many.

Proof that (-3)⁰ = 1
How to prove that a number to the zero power is one

Why is (-3)⁰ = 1? How is that proved?

Just like in the lesson about negative and zero exponents, you can look at the following sequence and ask what logically would come next:

(-3)⁴ = 81
(-3)³ = -27
(-3)² = 9
(-3)¹ = -3
(-3)⁰ = ????

You can present the same pattern for other numbers, too. Once your child discovers the rule for this sequence is division by -3 at each step, then the next logical step is that (-3)⁰ = 1.

The video below shows this same idea: teaching zero exponent starting with a pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof. The video also shows the idea for proof, explained below: we can multiply powers of the same base, and conclude from that what a number to zeroth power must be.

Zero Exponent video

The other proof idea is to first notice the following rule about multiplication (n is any integer):

n³ × n⁴ = (n×n×n) × (n×n×n×n) = n⁷

n⁶ × n² = (n×n×n×n×n×n) × (n×n) = n⁸

Can you notice the shortcut? For any whole number exponents x and y
you can just add the exponents:

n^x × n^y = (n×n×n ×...×n×n×n) × (n×...×n) = n^{x + y}

Mathematics is logical and the mathematical rules work in (nearly) all cases. You have noticed how many times a rule is stated to apply 'for any integer n' or for 'all whole numbers'. That makes mathematics beautiful. So suppose we don't know what (-3)⁰ is. Whatever (-3)⁰ is, if it obeys this rule above, then

(-3)⁷ × (-3)⁰ = (-3)^{7 + 0}

In other words,

(-3)⁷ × (-3)⁰ = (-3)⁷

(-3)³ × (-3)⁰ = (-3)^{3 + 0}

In other words,

(-3)³ × (-3)⁰ = (-3)³

(-3)¹⁵ × (-3)⁰ = (-3)^{15 +
0}

In other words,

(-3)¹⁵ × (-3)⁰ = (-3)¹⁵

...and so on for all kinds of possible exponents.

( In fact, we could write that (-3)^x × (-3)⁰ = (-3)^x, where x is any whole number. That is the way you see it written in some math books, because whenever something is true for all kinds of numbers, it is written using a letter to symbolize those 'all kinds of numbers' .)

Since we are supposing that we don't yet know what (-3)⁰ is, let's substitute P for it. Now look at the equations we found above. Knowing what you know about properties of multiplication, what kind of number can P be?

(-3)⁷ × P = (-3)⁷

(-3)³ × P = (-3)³

(-3)¹⁵ × P = (-3)¹⁵

So... What is the only number that when you multiply by it, nothing changes?

Question. What is the difference between -1 to the zero power and (-1) to the zero power? Will the answer = 1 for both?

Example 1: -1⁰ = ____ Example 2: (-1)⁰ = ___

Answer: As already explained, the answer to (-1)⁰ is 1 since we are raising the number -1 (negative 1) to the power zero. However, in the case of -1⁰, the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows. So in that case we first calculate 1⁰, and then take the opposite of that, which would result in -1.

Another example: in the expression -(-3)² , the first negative sign means you need to take the opposite number of whatever the rest of the expression comes to. So since (-3)² = 9, then -(-3)² = -9.

Question. Why is zero with a zero expoent come up with an error?? Please explain why it doesn't exist. In other words, what is 0⁰?

Answer:Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.

Since x⁰ is 1 for all numbers x other than 0, it would be logical to define that 0⁰ = 1.

But we could think of 0⁰ also having the value 0, because zero to any power (other than the zero power) is zero.

Also, the logarithm of 0⁰ would be 0 × infinity, which is in itself an indeterminate form. So laws of logs wouldn't work with it.

So because of these kind of problems, it is often left "indeterminate".

However, if zero to zeroth power needs to be defined as having some value, 1 is the most logical definition for its value. This can be "handy" if it makes some law to work (such as binomial theorem, or in combinatorial mathematics).

See also these articles from Dr. Math:

n^0 Power = 1: Defined or Proved?
Why are Operations of Zero so Strange?
What is 0 to the 0 power?

What is the difference between power and the exponent?
Varthan

Power is the whole thing, or the answer. Exponent is the little number. For example, 8 is a power (of 2) since 2³ = 8. 3 is the exponent. See also Powers and exponents at FactMonster.com.

Math Lessons menu

Place Value Ideas Teaching tens and ones Tens and ones practice Counting in groups of ten Skip-counting (0-100) Comparing (0-100) Cents and dimes Place Value - hundreds Comparing (0-1000) Place value - thousands Comparing thousands Rounding Place value - big numbers	Add/Subtract lessons Missing addend concept (0-10) Addition facts with 6 Addition & subtraction connection Within 0-100 Add/subtract whole tens (0-100) Easy principle of adding ones Fact families for basic facts Sum over next whole ten Add 2-digit numbers mentally Subtract 2-digit numbers mentally Carrying to tens Borrowing from tens Within 0-1000 Rounding to nearest ten Carry two times Borrow from hundreds Borrow two times Borrow over zero tens	Multiplication & Division Multiplication Multiplication concept as repeated addition Multiplication on number line Commutative Multiply by zero Word problems Order of operations Oral drilling guide Drilling tables of 2, 3, 5, or 10 Drilling tables of 4, 11, 9 Multiplying by whole tens Multiplying by 100 Distributive property Multiplying in columns the easy way Multiplying in columns Estimation when multiplying Division Division as making groups Division/multiplication connection Division is repeated subtraction Zero and one in division Not exact division/remainder Divisibility Long division as repeated subtraction Long division and why it works Remainder & long division Long division 0-10,000 Two-digit divisor Divisibility within 0-1000 Divisibility rules Factoring Sieve of Eratosthenes
Fraction Lessons Understanding fractions Part of whole group Mixed numbers Mixed number to fraction Fraction to mixed number Adding like fractions Equivalent fractions Adding unlike fractions Adding mixed numbers Subtracting mixed numbers Subtracting mixed numbers 2 Part of whole group 2 Measuring in inches Comparing fractions Simplifying fractions Whole number times a fraction Fraction times a fraction Multiplication as an area Simplify before multiplying Division of fractions - explain why it works Divide fractions by whole number Dividing fractions by fractions Dividing mixed numbers	Geometry Lessons Lines, rays, and angles Measuring angles Angles with the same vertex Parallel & perpendicular Triangles Altitude of a triangle Angles in a triangle Equilateral & isosceles triangles Circles Compass ruler constructions Symmetry Polygons Area of rectangles Area of right triangles Area of parallelograms Area of triangles	Decimals Lessons Tenths Tenths and place value Add decimals with tenths Multiply decimals with tenths Hundredths Hundredths as a place value Add decimals with hundredths Add decimals with hundredths 2 Multiply with hundredths Money problems. Comparing decimals

24K Gold Music Shows - 50s 60s oldies music show Performing 100s of songs of the popular artists from the oldies era. They are my favorites! Check them out! ~ Maria Miller

Proof that (-3)0 = 1 How to prove that a number to the zero power is one

Proof that (-3)⁰ = 1
How to prove that a number to the zero power is one