# 0 to the power of 0

= Notice that Sign up, Existing user? *See complete details for Better Score Guarantee. This is also the same reason why anything else raised to the power of 0 is 1. c On the other hand, other sources/branches of mathematics define 00=1.0^0 = 1.00=1. is In The Power of Zero, McKnight provides a concise, step-by-step roadmap on how to get to the 0% tax bracket by the time you retire, effectively eliminating tax rate risk from your retirement picture. We must define x^0=1 for all x , if the binomial theorem is to be valid when x=0 , y=0 , and/or x=-y . of exponents lim⁡x→0+x1ln⁡x? can be equal to In order for this to hold for x=0x = 0x=0 and n=1 n = 1n=1, we need 00=10^0 = 100=1. = 2 However, in the case of -1 0, the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows. The statement 00=10^0 = 100=1 is ambiguous and has been long debated in mathematics. Some believe it should be defined as 1 while others think it is 0, and some believe it is undefined. = Example 3: 000^000 represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1. . But 0 to the -5th power is 1 over 0 which is undefined and same with 0 to the -100th power. This is the third index law and is known as the Power of Zero. If it is convenient for the binomial theorem to assume 00=1,0^0=1,00=1, that is fine. 0 New user? Many sources consider 000^000 to be an "indeterminate form," or say that 000^000 is "undefined." For this reason, mathematicians say that = 0 Varsity Tutors connects learners with experts. Varsity Tutors does not have affiliation with universities mentioned on its website. Most of the arguments for why defining 00=10^0=100=1 is useful surround the fact that in some formulas, 00=10^0=100=1 makes the formula true for special cases involving 0. Rebuttal: Why do some problems on Brilliant say that 000^000 is undefined? × Varsity Tutors © 2007 - 2020 All Rights Reserved, Product b = Some examples of zero raised by positive powers. Year 8 Interactive Maths - Second Edition. On the other hand, This is highly debated. By contrast, the function 0^x is quite unimportant. , or Learn more in our Algebra Fundamentals course, built by experts for you. Zero to the Power of Zero What is 0 0 ?On one hand, any other number to the power of 0 is 1 (that's the Zero Exponent Property ).On the other hand, 0 to the power of anything else is 0 , because no matter how many times you multiply nothing by nothing, you still have nothing. In other words, what is 0 0? The exponent $0$ provides $0$ power (i.e. 0 by nothing, you still have nothing. still be true! 0 Instructors are independent contractors who tailor their services to each client, using their own style, As of 4/27/18. Example 9. While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions, or for the results obtained from the use of this information. -0.2 to the power of -2-16 to the power of -8-0.3 to the power of 3-0.6 to the power of 8; Disclaimer. Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. 2 of Powers Property, CCNA Collaboration - Cisco Certified Network Associate-Collaboration Courses & Classes, CCNA Service Provider - Cisco Certified Network Associate-Service Provider Courses & Classes, CTP - Certified Treasury Professional Test Prep, CRISC - Certified in Risk and Information Systems Control Test Prep, MCSE - Microsoft Certified Solutions Expert Courses & Classes, CIA - Certified Internal Auditor Test Prep. We can't have it both ways. On the other hand, if a mathematician who works with limits chooses to leave 000^000 as undefined, that is fine too! The exponent $1$ 'gives the number $1$ the power to transform into $3$. 0 2   Reply: Mathematics is a subject built upon definitions--there is no "universal truth" of what 000^000 really equals. ( Answer: As already explained, the answer to (-1) 0 is 1 since we are raising the number -1 (negative 1) to the power zero. 0 For example, lim⁡x→0e−1∣x∣=lim⁡x→0∣x∣=0,\lim_{x \to 0} e^{-\frac{1}{|x|}} = \lim_{x \to 0} |x| = 0,limx→0​e−∣x∣1​=limx→0​∣x∣=0, but lim⁡x→0(e−1∣x∣)∣x∣=e−1∣x∣⋅∣x∣=e−1.\lim_{x \to 0} \left( e^{-\frac{1}{|x|}} \right)^{|x|} = e^{-\frac{1}{|x|} \cdot |x|} = e^{-1}.x→0lim​(e−∣x∣1​)∣x∣=e−∣x∣1​⋅∣x∣=e−1.   Some people claim that 00=1 0 ^ 0 = 1 00=1. 7 gives no power of transformation), so $3^0$ gives no power of transformation to the number $1$, so $3^0=1$. , or ) The Power of Zero, the third index law. \Large \lim_{ x \rightarrow 0^+ } x ^{^ { \frac{ 1}{\ln x} }}?x→0+lim​xlnx1​? The Power of Zero, the third index law. Consider a to the power b and ask what happens as a and b both approach 0. So we first calculate 1 0, and then take the opposite of that, which would result in -1. 0 Log in. = Forgot password? Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. + But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. So this says, 0 In short, an exponent 'transforms' the number $1$, so $3$ (or $3^1$). , This lesson will go into the rule in more detail, explaining how it works and giving some examples.   0 Some of the arguments for why 000^000 is indeterminate or undefined are as follows: Argument 1: We know that a0=1a^0 = 1a0=1 (((for all a≠0),a \ne 0),a​=0), but 0a=00^a = 00a=0 (((for all a>0).a>0).a>0). Argument 2: With respect to limits, if lim⁡x→af(x)=lim⁡x→ag(x)=0\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0limx→a​f(x)=limx→a​g(x)=0, then lim⁡x→af(x)g(x)\lim_{x \to a} f(x)^{g(x)}limx→a​f(x)g(x) doesn't necessarily tend to any particular value. On one hand, any other number to the power of 0 a 2 ). But 0 to the -5th power is 1 over 0 which is undefined and same with 0 to the -100th power. Example 9. Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 have different limiting values when x decreases to 0. is     Note that, certainly, 00≠0.0^0 \ne 0.00​=0. Certainly, 0−10^{-1}0−1 does not equal 0, since we cannot divide by 0.