what is e to the power of 0

What is due east⁰? If you remember your exponents, the answer to this question is easy. For all numbers, raising that number to the 0th power is equal to one. Then we know that:

eastward⁰=1

This reply relies on an intrinsic property of the way exponentiation is defined. Exponentiation is defined every bit iterative multiplication, so the expression 10^north means you multiply x past itself n times. Therefore, any expression in the form x⁰ has a meaning of multiplying ten by itself 0 times. Multiplying a number past itself 0 times should return the same chemical element, which is the same every bit multiplying the element by i, the multiplicative identity. So, for all numbers ten, x⁰ should give y'all the multiplicative identity, which is equal to 1 (except when x=0, which is a special instance we volition consider later).

"A zero itself is nothing, but without a zilch you cannot count annihilation; therefore, a nix is something, all the same zero." — Dalai Lama

The to a higher place reasoning is based on the definition of the exponentiation performance. Let's take a await at the general definition of exponentiation and how we can reason to the claim that 10⁰=i.

What Is Exponentiation?

Exponentiation is a mathematical operation that involves two numbers, abase of operations b and anexponent northward. Ifn is a positive number, than exponentiation correspond to iterative multiplication of the base. So an exponential expression means y'all multiply northward copies of the base b together. In other words, the expression

b^north

means the product of multiplyingnorthbases b together. We tin see how this works by plugging in bodily numbers instead of variables. For instance, two³ can be rewritten every bit (ii×ii×ii)=viii. In other words, 2³ is just equal to the number two multiplied by itself three times. As well, 4⁴ is equal to (four×4×4×iv)=256.

Conversely, ifdue northis a negative number, so exponentiation defined as:

b^-north = 1/bⁿ

Exponentiation with a negative exponent -ncorresponds to the reciprocal ofbⁿ. This item definition of negative exponentiation is a consequence of a useful rule for combining exponents that we volition expect at in a bit. For now though, just recollect the two definitions of exponentiation by positive and negative integers:

b^northward=(b_one ×b_ii ×b_iii ×….×b_n )

b^-n =1/bⁿ

Exponentiation tin can exist seen as the inverse of the logarithm operation

Concept Check!

Take a expect at these problems involving exponentiation to make sure you empathise the basic concept. For each problem, endeavor to rewrite the problem every bit an extended series of iterative multiplication:

1. 5⁶
2. 7^-2
3. i^-1

Solutions

1. 5⁶= (5×five×5×5×five×v)
2. 7^-2= 1/(7×vii)
3. 1^-1 = one/1¹

Focusing on problem 3 shows us an interesting fact regarding the number 1 and exponentiation. Namely, for whatsoever exponentnorth, iⁿ=1. This is piece of cake to bear witness. Since exponentiation is divers as iterative multiplication, 1ⁿ just means "multiplyn copies of one." Yet, the product of one and 1 is e'er equal to 1, no matter how many 1s you multiply. And so it does not thing what nis; 1^{due north} will ever equal ane.

Nosotros tin show that 1ⁿ=i holds for negative exponents likewise. Remember thatb^-nis defined every bit i/b^{due north}.Therefore, 1^-northward must be equal to ane/1ⁿ. Since we just proved that iⁿ=i, one ^-nwill always equal one/1, which is but equal to 1. Therefore, for alln,one^-n=i.

Extra practice:See if you can prove that for all bases b,b ¹=b. Yous should be able to do this simply past reasoning about the meaning of the exponent performance.

"When yous brand yourself into zero, your ability becomes invincible." — Mahatma Ghandi

Arithmetic Operations Involving Exponents

The definition of the exponentiation functioning gives us an algorithmic fashion to deal with bug involving the multiplication or partitioning of terms with exponents. We will get-go await at the example of multiplying exponent terms.

Multiplication

In general, the rule for multiplying terms with exponents is:

bⁿ×b^m= b^n+thou

We can show that this rule must exist true with a elementary numerical instance. Say we have 2⁴ and 2^three.  2^iv is the aforementioned as (2×ii×2×2) and 2ⁱⁱⁱ is (2×2×ii), then 2⁴×2³ = (2×2×2×2)(2×2×2). Since multiplication is associative (it does not affair where we put the parentheses) we tin combine all the terms on the correct side of this equation into i large parenthetical expression:

2⁴×2^three=(two×2×2×2×2×2×ii)=2^three+four=ii⁷

We can generalize this reasoning for any baseb, and whatsoever exponents thouanddue north. By definition of the exponentiation operation nosotros know that:

b ^ane =b,

and that

b^{due north+ane} = bⁿ×b ¹

Generalizing this case past settingb ⁱ=b^m gives us our final expression: bⁿ×b^m= bⁿ⁺¹⁰⁰⁰

Sectionalisation

In that location are also rules for dividing exponentiated terms. In full general, the rule is:

bⁿ÷b^m= b^n−m

Once over again, we tin can bear witness this dominion to be true for a specific numerical case, say two⁵ and 2^two. ii^vis the same as (2×two×2×2×ii) and ii² is (2×2). So 2^v÷2² is the aforementioned as (2×2×2×2×two)/(2×ii). Since nosotros are dividing, nosotros can cancel out pairs of like terms and get rid of them, which only leaves (2×2×2).

Incidentally, this particular rule of dividing exponentiated terms is what gives united states our definitions of exponentiation by 0.

Exponentiation

There are too rules for raising exponential terms themselves to exponents. In full general, raising an exponent term to an exponent is:

(b^north ) ^g= b^n×m

This particular definition of nested exponentiation is a consequence of the associativity of multiplication. terms. The expression (b^north ) ^m  tells united states to multiply togetherone thousand copies of the base of operationsbⁿ.The base,b^northwardtells us to multiplynorthwardcopies of the baseb. So all together, we would havemcopies ofbⁿwhere eachbⁿhasnorthwardcopies ofb. To figure out the total number ofbsouthward multiplied together, we tin can just multiplymby northward torequite united states of america our final exponent expression.

Here is a numerical case to illustrate the betoken. Say we have (ii²)^three. This expression tells us to multiply together 3 copies of the base 2². This is the same as (two×two)×(2×two)×(two×2). Since multiplication is associative, we can get rid of the parentheses and combine all the terms into one expression (two×2×2×two×2×2) = (twoⁱⁱ)ⁱⁱⁱ = 2^six.

Exponentiation By 0

Now that we have some exponent rules under our belt, we can see exactly why raising any number to the 0th ability ever equals 1. Recall that when dividing exponentiated terms, the general rule is:

b^{due north}÷b^grand= b^n−m

This is equivalent tob^north/b^chiliad.If we set bothnorthward andthousand equal to i we get:

b¹÷b¹ = b¹⁻¹= b^ane/bⁱ = b⁰

Sinceb¹ is equal to just band any number divided by itself is equal to ane, information technology follows necessarily that

b ⁰=b¹/b¹= b/b = 1

This expression is true for any not-zero numberb. Some other way of maxim this is thatb ⁰ corresponds to dividingb past itself, which will always equal 1.

Exponentiation Past Negative Numbers

At present that we have proved that exponentiation past 0 ever equals 1, we can generate the definition of negative exponentiation hinted at earlier. Recall that for negative exponentiation:

b^-n =1/bⁿ

According to the definition of exponentiation, nosotros know that:

bⁿ = (b ^{due north+i})/b

The to a higher place statement simply follows from what information technology means to apply the exponentiation operator. We also know that this statement is true in the example thatdue north=0:

b⁰=b ¹/b = i

Now, extending this definition to n =-1 gives us:

b ^-i =b ⁰/b = 1/b

We tin can extend this reasoning to all negativento get in at our identity statement for negative exponentiation:

b^-n = 1/b^northward

Raising 0 To A Ability

All the to a higher place rules are defined with a not-zero base b. What happens if nosotros permitb =0? The operation of raising zero to an exponent has some special rules.

First, when due north is positive, 0ⁿ=0. This is easy enough to see; the expression 0ⁿ ways "multiply togethern copies of 0." 0×0 is e'er equal to 0 no matter how many copies of 0 yous accept, and so it follows that for all positiven,0 ^north = 0.

Whendue northis negative, 0 ⁿ is undefined. The reason why is that raising 0 to a negative exponent implies division by 0, which is undefined. Think of it this style, we know that for allb andn,b^-n = 1/bⁿ.If we letb= 0, so we get:

0^-north = i/0ⁿ = 1/0.

Division by zippo is an undefined operation, and so raising 0 to some negative power is also an undefined performance.

0 To The 0th Power

What well-nigh when bothb andn are 0, so 0⁰? Here we commencement to go to the controversial territory. Currently, there is no universally accepted value for the expression 0⁰. Depending on the field, mathematicians will give you different answers. In bones algebra, 0⁰ is normally only defined as equal to 1, notwithstanding, this is often for reasons of simplicity rather than logical rigor.

"Zero-zero is a large score." — Ron Atkinson

Some mathematicians contend that 0⁰ is undefined for the same reason that raising 0 to a negative exponent is undefined; they both imply a segmentation past 0, which is an undefined operation in algebra. Following the normal rules for non-nothing exponents while substituting 0 seems to betoken that the expression 0⁰ is the aforementioned every bit 0/0.

Other mathematicians argue on this ground that 0⁰ is not undefined, justindeterminate. The reason why is that the expression 0/0 does non give us enough information to determine its value. Expressions like 0/0 are unremarkably seen in the context of calculus, as i takes the ratio of two derivatives every bit they approach their limits. In these cases, an answer of 0/0 does not mean that the respond is undefined, merely that nosotros need more than information to determine its value.

Well-nigh Alex Bolano PRO INVESTOR

When Alex isn't nerdily stalking the cyberspace for science news, he enjoys tabletop RPGs and making really obscure TV references. Alex has a Masters'south degree from the Academy of Missouri-St. Louis.

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