Answer :
Sure, let's convert the given expression \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\) into a radical form.
Step-by-step solution:
1. Identify the base and exponent:
- The expression given is \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\).
- Here, \(\frac{17}{19}\) is the base.
- \(\frac{17}{23}\) is the exponent.
2. Understand the property of exponents related to radicals:
- The expression \(a^{b/c}\) can be written in radical form as \(\sqrt[c]{a^b}\).
- Applying this property to \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\), we can say:
[tex]\[ \left(\frac{17}{19}\right)^{\frac{17}{23}} = \sqrt[23]{\left(\frac{17}{19}\right)^{17}} \][/tex]
3. Calculate the numerical approximation:
- The given base \(\frac{17}{19}\) approximately equals \(0.8947368421052632\).
- Calculating \((0.8947368421052632)^{17}\) results in approximately \(0.15094559624221696\).
4. Express the radical form with the known values:
- Therefore, \(\sqrt[23]{(0.8947368421052632)^{17}}\) represents the expression in radical form.
To summarize, if we start with \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\), we can write it in radical form as:
[tex]\[ \sqrt[23]{\left(\frac{17}{19}\right)^{17}} \quad \text{or} \quad 23\sqrt{0.8947368421052632^{17}} \][/tex]
Step-by-step solution:
1. Identify the base and exponent:
- The expression given is \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\).
- Here, \(\frac{17}{19}\) is the base.
- \(\frac{17}{23}\) is the exponent.
2. Understand the property of exponents related to radicals:
- The expression \(a^{b/c}\) can be written in radical form as \(\sqrt[c]{a^b}\).
- Applying this property to \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\), we can say:
[tex]\[ \left(\frac{17}{19}\right)^{\frac{17}{23}} = \sqrt[23]{\left(\frac{17}{19}\right)^{17}} \][/tex]
3. Calculate the numerical approximation:
- The given base \(\frac{17}{19}\) approximately equals \(0.8947368421052632\).
- Calculating \((0.8947368421052632)^{17}\) results in approximately \(0.15094559624221696\).
4. Express the radical form with the known values:
- Therefore, \(\sqrt[23]{(0.8947368421052632)^{17}}\) represents the expression in radical form.
To summarize, if we start with \(\left(\frac{17}{19}\right)^{\frac{17}{23}}\), we can write it in radical form as:
[tex]\[ \sqrt[23]{\left(\frac{17}{19}\right)^{17}} \quad \text{or} \quad 23\sqrt{0.8947368421052632^{17}} \][/tex]
