Sure, let's rewrite each expression step by step.
### Part A: [tex]\( x \sqrt[7]{x^2} \)[/tex]
1. Start by rewriting the radical expression using exponents:
[tex]\[ \sqrt[7]{x^2} = x^{2/7} \][/tex]
2. Multiply this by [tex]\( x \)[/tex]:
[tex]\[ x \cdot x^{2/7} \][/tex]
3. Add the exponents (since the bases are the same):
[tex]\[ x^{1 + 2/7} = x^{9/7} \][/tex]
So, the rewritten expression for Part A is:
[tex]\[ x \sqrt[7]{x^2} = x^{9/7} \][/tex]
### Part B: [tex]\( \sqrt[9]{x^7} \)[/tex]
1. Rewrite the radical expression using exponents:
[tex]\[ \sqrt[9]{x^7} = x^{7/9} \][/tex]
So, the rewritten expression for Part B is:
[tex]\[ \sqrt[9]{x^7} = x^{7/9} \][/tex]
### Part C: [tex]\( x \sqrt[7]{x} \)[/tex]
1. Rewrite the radical expression using exponents:
[tex]\[ \sqrt[7]{x} = x^{1/7} \][/tex]
2. Multiply this by [tex]\( x \)[/tex]:
[tex]\[ x \cdot x^{1/7} \][/tex]
3. Add the exponents:
[tex]\[ x^{1 + 1/7} = x^{8/7} \][/tex]
So, the rewritten expression for Part C is:
[tex]\[ x \sqrt[7]{x} = x^{8/7} \][/tex]
### Part D: [tex]\( \left(\frac{1}{\sqrt[7]{x}}\right)^9 \)[/tex]
1. Rewrite the radical expression using exponents:
[tex]\[ \frac{1}{\sqrt[7]{x}} = x^{-1/7} \][/tex]
2. Raise this to the 9th power:
[tex]\[ \left( x^{-1/7} \right)^9 \][/tex]
3. Multiply the exponents:
[tex]\[ x^{-1/7 \cdot 9} = x^{-9/7} \][/tex]
So, the rewritten expression for Part D is:
[tex]\[ \left(\frac{1}{\sqrt[7]{x}}\right)^9 = x^{-9/7} \][/tex]
In summary:
A. [tex]\( x \sqrt[7]{x^2} = x^{9/7} \)[/tex]
B. [tex]\( \sqrt[9]{x^7} = x^{7/9} \)[/tex]
C. [tex]\( x \sqrt[7]{x} = x^{8/7} \)[/tex]
D. [tex]\( \left(\frac{1}{\sqrt[7]{x}}\right)^9 = x^{-9/7} \)[/tex]
