Question

do you understand?
1.
what do the zero
exponent and negative exponent properties
mean?

Explanation:

• Zero Exponent Property: For any non - zero real number \(a\), \(a^{0}=1\). This can be understood from the quotient rule of exponents. The quotient rule states that \(\frac{a^{m}}{a^{n}}=a^{m - n}\) (where \(a

eq0\)). If \(m = n\), then \(\frac{a^{m}}{a^{m}}=a^{m - m}=a^{0}\), and since \(\frac{a^{m}}{a^{m}} = 1\) (as long as \(a
eq0\), because a non - zero number divided by itself is 1), we have \(a^{0}=1\) for \(a
eq0\).

• Negative Exponent Property: For any non - zero real number \(a\) and any integer \(n\), \(a^{-n}=\frac{1}{a^{n}}\) (or equivalently \(\frac{1}{a^{-n}}=a^{n}\)). Using the quotient rule again, \(\frac{a^{m}}{a^{n}}=a^{m - n}\). If \(m = 0\) and \(n>0\), then \(\frac{a^{0}}{a^{n}}=a^{0 - n}=a^{-n}\). But we know \(a^{0}=1\) (for \(a

eq0\)), so \(\frac{1}{a^{n}}=a^{-n}\) (when \(a
eq0\)).

Answer:

• Zero Exponent Property: For a non - zero real number \(a\), \(a^{0}=1\) (e.g., \(5^{0}=1\), \((-3)^{0}=1\) but \(0^{0}\) is undefined).
• Negative Exponent Property: For a non - zero real number \(a\) and integer \(n\), \(a^{-n}=\frac{1}{a^{n}}\) (e.g., \(2^{-3}=\frac{1}{2^{3}}=\frac{1}{8}\), \((\frac{1}{3})^{-2}=3^{2} = 9\)).