We need to identify which of these statements correctly apply the negative exponent property. The negative exponent property states that [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex].
Let's evaluate each statement individually:
1. [tex]\(17^{-\frac{1}{4}} = \frac{1}{17^{\frac{1}{4}}}\)[/tex]
- This is a correct application of the negative exponent property. For any positive number [tex]\(a\)[/tex] and any positive real number [tex]\(b\)[/tex], [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex]. Thus, [tex]\(17^{-\frac{1}{4}} = \frac{1}{17^{\frac{1}{4}}}\)[/tex].
2. [tex]\(6^{-\frac{1}{3}} = -6^{\frac{1}{3}}\)[/tex]
- This statement is incorrect. According to the negative exponent property, it should be [tex]\(6^{-\frac{1}{3}} = \frac{1}{6^{\frac{1}{3}}}\)[/tex], not [tex]\(-6^{\frac{1}{3}}\)[/tex].
3. [tex]\(y^{\frac{1}{2}} = \frac{1}{y^{\frac{1}{2}}}\)[/tex]
- This statement is incorrect. The expression [tex]\(y^{\frac{1}{2}}\)[/tex] represents the square root of [tex]\(y\)[/tex]. The correct application of the negative exponent property would be [tex]\(\frac{1}{y^{\frac{1}{2}}}\)[/tex] or [tex]\(y^{-\frac{1}{2}} = \frac{1}{y^{\frac{1}{2}}}\)[/tex].
4. [tex]\(8^{-\frac{1}{6}} = -\frac{1}{8^{\frac{1}{8}}}\)[/tex]
- This statement is incorrect. According to the negative exponent property, it should be [tex]\(8^{-\frac{1}{6}} = \frac{1}{8^{\frac{1}{6}}}\)[/tex], not [tex]\(-\frac{1}{8^{\frac{1}{8}}}\)[/tex].
5. [tex]\(x^{-\frac{1}{7}} = \frac{x}{x^{\frac{3}{7}}}\)[/tex]
- This statement is incorrect. The correct application of the negative exponent property would produce [tex]\(x^{-\frac{1}{7}} = \frac{1}{x^{\frac{1}{7}}}\)[/tex].
To summarize, the two statements that correctly apply the negative exponent property are:
- [tex]\(17^{-\frac{1}{4}}=\frac{1}{17^{\frac{1}{4}}}\)[/tex]
Thus, the correct answer is:
[tex]\(\bigcirc \, 17^{-\frac{1}{4}}=\frac{1}{17^{\frac{1}{4}}}\)[/tex]
