To solve the expression [tex]\( E = \left[\sqrt[19]{19} \sqrt[19]{19} \cdot \sqrt[19]{19^{-1}}-\sqrt[19]{19}\right]^{\frac{19}{19} 19} \)[/tex], we can follow these steps:
1. Simplify the terms inside the brackets:
- The expression starts with [tex]\(\sqrt[19]{19} \sqrt[19]{19} \cdot \sqrt[19]{19^{-1}}\)[/tex].
2. Combine the exponents using the properties of exponents:
- Recall that [tex]\(\sqrt[19]{19}\)[/tex] can also be written as [tex]\(19^{\frac{1}{19}}\)[/tex].
- Therefore, [tex]\(\sqrt[19]{19} \sqrt[19]{19}\)[/tex] becomes:
[tex]\[
19^{\frac{1}{19}} \cdot 19^{\frac{1}{19}}
\][/tex]
- Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] we combine the exponents:
[tex]\[
19^{\frac{1}{19} + \frac{1}{19}} = 19^{\frac{2}{19}}
\][/tex]
3. Simplify the multiplication with [tex]\(\sqrt[19]{19^{-1}}\)[/tex]:
- Recall that [tex]\(\sqrt[19]{19^{-1}}\)[/tex] can be written as [tex]\(19^{-\frac{1}{19}}\)[/tex].
- Therefore, we have:
[tex]\[
19^{\frac{2}{19}} \cdot 19^{-\frac{1}{19}}
\][/tex]
- Again, using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we combine the exponents:
[tex]\[
19^{\frac{2}{19} + (-\frac{1}{19})} = 19^{\frac{2}{19} - \frac{1}{19}} = 19^{\frac{1}{19}}
\][/tex]
4. Subtract [tex]\(\sqrt[19]{19}\)[/tex]:
- Now we have:
[tex]\[
19^{\frac{1}{19}} - 19^{\frac{1}{19}}
\][/tex]
- Clearly, subtracting the same quantity from itself results in 0:
[tex]\[
19^{\frac{1}{19}} - 19^{\frac{1}{19}} = 0
\][/tex]
5. Raise the result to the given power:
- The expression now simplifies to:
[tex]\[
\left[0\right]^{\frac{19}{19} \cdot 19}
\][/tex]
- Simplify the exponent [tex]\(\frac{19}{19} \cdot 19\)[/tex]:
[tex]\[
\frac{19}{19} \cdot 19 = 1 \cdot 19 = 19
\][/tex]
6. Apply the exponent to the 0 inside the brackets:
- Any number raised to the power of 0 is 0:
[tex]\[
0^{19} = 0
\][/tex]
Thus, the final result is:
[tex]\[
E = 0
\][/tex]
