Every number in Mathematics can be written to the power (or exponent) of one (1). Consider the examples below:
\(*\) \(2 = 2^1\)
\(*\) \(a = a^1\)
\(*\) \(200 = 200^1\) etc.
We also know that every number divided by itself is one (1). Consider the examples below:
\(*\) \(\frac{2}{2} = 1\)
\(*\) \(\frac{a}{a} = 1\)
\(*\) \(\frac{100}{100} = 1\)
Also, in indices, we have learnt that;
\((i)\) \(a^x \div a^y\)
\(\Rightarrow \frac{a^x}{a^y}\)
\(\Rightarrow a^{x-y}\)
The number one (1) can be expressed as a fraction in terms of any number. The following are all equivalent fractions to 1.
\(\hspace{0.5cm} 1 = \frac{2}{2}\)
\(\hspace{0.5cm} 1 = \frac{3}{3}\)
\(\hspace{0.5cm} 1 = \frac{4}{4}\)
\(\hspace{0.5cm} 1 = \frac{5}{5}\)
\(\hspace{0.5cm} 1 = \frac{x}{x} \)
From the definitions above, it follows that;
\(\Rightarrow\) \(\frac{x}{x} = 1\)
\(\Rightarrow\) \(\frac{x^1}{x^1} = 1\)
\(\Rightarrow\) \(x^{1-1} = 1\)
\(\Rightarrow\) \(x^0 = 1\)
\(\therefore\) Every number raised to the power (or exponent) of zero (0) is equal to 1.
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