Question

select all the expressions that are equivalent to ((10^{-2})^0). (\frac{1}{10^{-2}}) (\frac{1}{10^{0}}) (1) (10)

Explanation:

Step1: Apply the zero - exponent rule

The zero - exponent rule states that for any non - zero number \(a\), \(a^{0}=1\). In the expression \((10^{-2})^{0}\), since the base \(10^{-2}\) is non - zero (because \(10^{-2}=\frac{1}{10^{2}}=\frac{1}{100}
eq0\)), by the zero - exponent rule, \((10^{-2})^{0} = 1\).

Step2: Analyze \(\frac{1}{10^{-2}}\)

Using the negative - exponent rule \(a^{-n}=\frac{1}{a^{n}}\) (or \(\frac{1}{a^{-n}}=a^{n}\)), for \(\frac{1}{10^{-2}}\), we have \(\frac{1}{10^{-2}}=10^{2}=100
eq1\), so \(\frac{1}{10^{-2}}\) is not equivalent to \((10^{-2})^{0}\).

Step3: Analyze \(\frac{1}{10^{0}}\)

Since \(10^{0} = 1\) (by the zero - exponent rule), then \(\frac{1}{10^{0}}=\frac{1}{1}=1\), so \(\frac{1}{10^{0}}\) is equivalent to \((10^{-2})^{0}\).

Step4: Analyze \(10\)

\(10
eq1\), so \(10\) is not equivalent to \((10^{-2})^{0}\).

Step5: Analyze \(1\)

We already found that \((10^{-2})^{0}=1\), so \(1\) is equivalent to \((10^{-2})^{0}\).

Answer:

The expressions equivalent to \((10^{-2})^{0}\) are \(\frac{1}{10^{0}}\) and \(1\) (i.e., the options \(\frac{1}{10^{0}}\) and \(1\)). The expressions \(\frac{1}{10^{-2}}\) and \(10\) are not equivalent. So the correct selections are \(\frac{1}{10^{0}}\) and \(1\).