From science to arts, IDNLearn.com has the answers to all your questions. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.

Which is equivalent to [tex]\sqrt[3]{8}^{\frac{1}{4} x}[/tex]?

A. [tex]8^{\frac{3}{4} x}[/tex]
B. [tex]\sqrt[7]{8}^x[/tex]
C. [tex]\sqrt[12]{8} x[/tex]
D. [tex]8^{\frac{3}{4 x}}[/tex]


Sagot :

To determine which expression is equivalent to [tex]\(\sqrt[3]{8}^{\frac{1}{4} x}\)[/tex], we need to simplify the given expression step-by-step.

1. First, let's rewrite [tex]\(\sqrt[3]{8}\)[/tex]:
[tex]\[ \sqrt[3]{8} = 8^{\frac{1}{3}} \][/tex]

2. Now substitute this result into the original expression:
[tex]\[ \sqrt[3]{8}^{\frac{1}{4} x} = \left(8^{\frac{1}{3}}\right)^{\frac{1}{4} x} \][/tex]

3. Use the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex] to simplify the expression:
[tex]\[ \left(8^{\frac{1}{3}}\right)^{\frac{1}{4} x} = 8^{(\frac{1}{3} \cdot \frac{1}{4} x)} = 8^{\frac{x}{12}} \][/tex]

Now, we need to compare this simplified form [tex]\(8^{\frac{x}{12}}\)[/tex] to the options given:

Option 1: [tex]\(8^{\frac{3}{4} x}\)[/tex]
[tex]\[ 8^{\frac{3}{4} x} \neq 8^{\frac{x}{12}} \][/tex]

Option 2: [tex]\(\sqrt[7]{8}^x\)[/tex]
[tex]\[ \sqrt[7]{8}^x = (8^{\frac{1}{7}})^x = 8^{\frac{x}{7}} \neq 8^{\frac{x}{12}} \][/tex]

Option 3: [tex]\(\sqrt[12]{8} x\)[/tex]
[tex]\[ \sqrt[12]{8} x = (8^{\frac{1}{12}}) x \neq 8^{\frac{x}{12}} \][/tex]

Option 4: [tex]\(8^{\frac{3}{4 x}}\)[/tex]
[tex]\[ 8^{\frac{3}{4 x}} \neq 8^{\frac{x}{12}} \][/tex]

None of the given options are equivalent to [tex]\(8^{\frac{x}{12}}\)[/tex].

Thus, without any errors, the equivalent expression to [tex]\(\sqrt[3]{8}^{\frac{1}{4} x}\)[/tex] is:
[tex]\[8^{\frac{x}{12}}\][/tex]

There appears to be an oversight or missing option in the list provided.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.