High School

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The lung volume for mammals can be modeled using the expression [tex]170 x^{\frac{4}{3}}[/tex], where [tex]x[/tex] is the mass of the mammal. How can this expression be written using radicals?

A) [tex]170 \sqrt[3]{x^4}[/tex]

B) [tex]170 \sqrt[4]{x^5}[/tex]

C) [tex]\sqrt[3]{170 x^4}[/tex]

D) [tex]\sqrt[4]{170 x^5}[/tex]

Answer :

Sure! Let's solve the problem step-by-step.

We have the expression for lung volume of mammals: [tex]\(170 x^{\frac{4}{3}}\)[/tex].

We need to express this using radicals.

1. Understanding the Power:
- The term [tex]\(x^{\frac{4}{3}}\)[/tex] means [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{4}{3}\)[/tex].

2. Converting to Radicals:
- The expression [tex]\(x^{\frac{4}{3}}\)[/tex] can be rewritten using the properties of exponents as [tex]\((x^4)^{\frac{1}{3}}\)[/tex].
- This represents the cube root of [tex]\(x^4\)[/tex].

3. Putting it Together:
- Therefore, the entire expression [tex]\(170 x^{\frac{4}{3}}\)[/tex] can be rewritten in radical form as:
[tex]\[ 170 \sqrt[3]{x^4} \][/tex]

Considering the options given, the correct choice is:
[tex]\[ \text{A) } 170 \sqrt[3]{x^4} \][/tex]

So, the expression for lung volume in terms of radicals is [tex]\(170 \sqrt[3]{x^4}\)[/tex].

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