Why is anything the power of 0 always 1?

Now I know this is one known fact that students just accept. Why is it that anything raised to the power of 0 is 1? We know that $2^{3}=2 \times 2 \times 2 = 8$ , meaning we take the 2 and multiply it three times. So wouldn’t $2^0$ mean we multiply 2 zeroth times, and get back 2? So thats baffling.

This can be solved intuitively, like how we looked at $0!$ previously. We can look at $b^a$ as the number of ways to arrange a set of $a$ number with numbers from $1$ to $b$ numbers.

Consider $2^3$ . This means we arrange the set of 3 numbers where the numbers are just 1 and 2. So we have $\{1, 1, 1 \}, \{1, 1, 2 \}, \{1, 2, 2 \}, \{2, 2, 2 \}, \{1, 2, 1 \}, \{2, 2, 1 \}, \{2, 1, 1 \}, \{2, 1, 2 \}$

And we find the number of ways to be eight!

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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May 1st, 2016 04:06 PM

Why is anything the power of 0 always 1?

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