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Sagot :
To rewrite the expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in its simplest radical form, follow these steps:
1. Simplify the exponent:
[tex]\[ -\frac{3}{6} = -\frac{1}{2} \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{1}{x^{-\frac{1}{2}}} \][/tex]
2. Rewrite using exponential rules:
Recall that [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. So, [tex]\(x^{-\frac{1}{2}}\)[/tex] is:
[tex]\[ \frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}} \][/tex]
3. Convert to radical form:
The exponent [tex]\(\frac{1}{2}\)[/tex] signifies the square root. So, [tex]\(x^{\frac{1}{2}}\)[/tex] can be written as:
[tex]\[ \sqrt{x} \][/tex]
Now, placing it all together, the original expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in simplest radical form is:
[tex]\[ \boxed{\sqrt{x}} \][/tex]
The step-by-step process yields the final simplified radical form of the given expression as [tex]\(\sqrt{x}\)[/tex].
1. Simplify the exponent:
[tex]\[ -\frac{3}{6} = -\frac{1}{2} \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{1}{x^{-\frac{1}{2}}} \][/tex]
2. Rewrite using exponential rules:
Recall that [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. So, [tex]\(x^{-\frac{1}{2}}\)[/tex] is:
[tex]\[ \frac{1}{x^{-\frac{1}{2}}} = x^{\frac{1}{2}} \][/tex]
3. Convert to radical form:
The exponent [tex]\(\frac{1}{2}\)[/tex] signifies the square root. So, [tex]\(x^{\frac{1}{2}}\)[/tex] can be written as:
[tex]\[ \sqrt{x} \][/tex]
Now, placing it all together, the original expression [tex]\(\frac{1}{x^{-\frac{3}{6}}}\)[/tex] in simplest radical form is:
[tex]\[ \boxed{\sqrt{x}} \][/tex]
The step-by-step process yields the final simplified radical form of the given expression as [tex]\(\sqrt{x}\)[/tex].
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