Answer :
To determine which of the given expressions are equivalent to [tex]\( x^{9/4} \)[/tex], we need to see if we can rewrite each expression to match [tex]\( x^{9/4} \)[/tex].
Let's analyze each expression step-by-step:
Expression A: [tex]\(\left(x^4\right)^{1 / 9}\)[/tex]
- Rewriting this expression using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ \left(x^4\right)^{1 / 9} = x^{4 \cdot (1 / 9)} = x^{4/9} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
Expression B: [tex]\((\sqrt[4]{x})^9\)[/tex]
- Rewriting [tex]\(\sqrt[4]{x}\)[/tex] as [tex]\( x^{1/4} \)[/tex], we get:
[tex]\[ (\sqrt[4]{x})^9 = (x^{1/4})^9 = x^{(1/4) \cdot 9} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression C: [tex]\(\sqrt[9]{x^4}\)[/tex]
- Rewriting [tex]\(\sqrt[9]{x^4}\)[/tex] as [tex]\( (x^4)^{1/9} \)[/tex], we get:
[tex]\[ \sqrt[9]{x^4}= (x^4)^{1/9} = x^{4 \cdot (1/9)} = x^{4/9} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
Expression D: [tex]\(\sqrt[4]{x^9}\)[/tex]
- Rewriting [tex]\(\sqrt[4]{x^9}\)[/tex] as [tex]\( (x^9)^{1/4} \)[/tex], we get:
[tex]\[ \sqrt[4]{x^9} = (x^9)^{1/4} = x^{9 \cdot (1/4)} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression E: [tex]\((x^9)^{1 / 4}\)[/tex]
- Rewriting [tex]\((x^9)^{1 / 4}\)[/tex] using the power of a power rule, we get:
[tex]\[ (x^9)^{1/4} = x^{9 \cdot (1/4)} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression F: [tex]\((\sqrt[3]{x})^4\)[/tex]
- Rewriting [tex]\(\sqrt[3]{x}\)[/tex] as [tex]\( x^{1/3} \)[/tex], we get:
[tex]\[ (\sqrt[3]{x})^4 = (x^{1/3})^4 = x^{(1/3) \cdot 4} = x^{4/3} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
So, the expressions that are equivalent to [tex]\( x^{9/4} \)[/tex] are:
- Expression B: [tex]\((\sqrt[4]{x})^9\)[/tex]
- Expression D: [tex]\(\sqrt[4]{x^9}\)[/tex]
- Expression E: [tex]\((x^9)^{1 / 4}\)[/tex]
Therefore, the correct answers are B, D, and E.
Let's analyze each expression step-by-step:
Expression A: [tex]\(\left(x^4\right)^{1 / 9}\)[/tex]
- Rewriting this expression using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ \left(x^4\right)^{1 / 9} = x^{4 \cdot (1 / 9)} = x^{4/9} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
Expression B: [tex]\((\sqrt[4]{x})^9\)[/tex]
- Rewriting [tex]\(\sqrt[4]{x}\)[/tex] as [tex]\( x^{1/4} \)[/tex], we get:
[tex]\[ (\sqrt[4]{x})^9 = (x^{1/4})^9 = x^{(1/4) \cdot 9} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression C: [tex]\(\sqrt[9]{x^4}\)[/tex]
- Rewriting [tex]\(\sqrt[9]{x^4}\)[/tex] as [tex]\( (x^4)^{1/9} \)[/tex], we get:
[tex]\[ \sqrt[9]{x^4}= (x^4)^{1/9} = x^{4 \cdot (1/9)} = x^{4/9} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
Expression D: [tex]\(\sqrt[4]{x^9}\)[/tex]
- Rewriting [tex]\(\sqrt[4]{x^9}\)[/tex] as [tex]\( (x^9)^{1/4} \)[/tex], we get:
[tex]\[ \sqrt[4]{x^9} = (x^9)^{1/4} = x^{9 \cdot (1/4)} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression E: [tex]\((x^9)^{1 / 4}\)[/tex]
- Rewriting [tex]\((x^9)^{1 / 4}\)[/tex] using the power of a power rule, we get:
[tex]\[ (x^9)^{1/4} = x^{9 \cdot (1/4)} = x^{9/4} \][/tex]
This is equivalent to [tex]\( x^{9/4} \)[/tex].
Expression F: [tex]\((\sqrt[3]{x})^4\)[/tex]
- Rewriting [tex]\(\sqrt[3]{x}\)[/tex] as [tex]\( x^{1/3} \)[/tex], we get:
[tex]\[ (\sqrt[3]{x})^4 = (x^{1/3})^4 = x^{(1/3) \cdot 4} = x^{4/3} \][/tex]
This is not equivalent to [tex]\( x^{9/4} \)[/tex].
So, the expressions that are equivalent to [tex]\( x^{9/4} \)[/tex] are:
- Expression B: [tex]\((\sqrt[4]{x})^9\)[/tex]
- Expression D: [tex]\(\sqrt[4]{x^9}\)[/tex]
- Expression E: [tex]\((x^9)^{1 / 4}\)[/tex]
Therefore, the correct answers are B, D, and E.
B and D and E
B = (c^1/4)^9 = x^9/4
D = (x^9*1/4) = x^9/4
E = (x^9)^1/4 = x^9/4
B = (c^1/4)^9 = x^9/4
D = (x^9*1/4) = x^9/4
E = (x^9)^1/4 = x^9/4
