Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To determine which exponential expression is equivalent to [tex]\((\sqrt[3]{t})^2\)[/tex], we need to simplify it step-by-step.
1. Rewrite the Radicals as Exponents:
The expression [tex]\(\sqrt[3]{t}\)[/tex] can be rewritten in exponential form. By definition, the cube root of [tex]\(t\)[/tex] is [tex]\(t\)[/tex] raised to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \sqrt[3]{t} = t^{\frac{1}{3}} \][/tex]
2. Apply the Power Rule for Exponents:
We need to raise this expression to the power of 2:
[tex]\[ (\sqrt[3]{t})^2 = \left(t^{\frac{1}{3}}\right)^2 \][/tex]
3. Simplify the Exponent:
When raising a power to another power, you multiply the exponents. Therefore, we multiply [tex]\(\frac{1}{3}\)[/tex] by 2:
[tex]\[ \left(t^{\frac{1}{3}}\right)^2 = t^{\frac{1}{3} \cdot 2} = t^{\frac{2}{3}} \][/tex]
The equivalent expression is [tex]\(t^{\frac{2}{3}}\)[/tex], which corresponds to option (B).
Thus, the correct answer is:
[tex]\[ \boxed{t^{\frac{2}{3}}} \][/tex]
Therefore, option (B) is the correct choice.
1. Rewrite the Radicals as Exponents:
The expression [tex]\(\sqrt[3]{t}\)[/tex] can be rewritten in exponential form. By definition, the cube root of [tex]\(t\)[/tex] is [tex]\(t\)[/tex] raised to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \sqrt[3]{t} = t^{\frac{1}{3}} \][/tex]
2. Apply the Power Rule for Exponents:
We need to raise this expression to the power of 2:
[tex]\[ (\sqrt[3]{t})^2 = \left(t^{\frac{1}{3}}\right)^2 \][/tex]
3. Simplify the Exponent:
When raising a power to another power, you multiply the exponents. Therefore, we multiply [tex]\(\frac{1}{3}\)[/tex] by 2:
[tex]\[ \left(t^{\frac{1}{3}}\right)^2 = t^{\frac{1}{3} \cdot 2} = t^{\frac{2}{3}} \][/tex]
The equivalent expression is [tex]\(t^{\frac{2}{3}}\)[/tex], which corresponds to option (B).
Thus, the correct answer is:
[tex]\[ \boxed{t^{\frac{2}{3}}} \][/tex]
Therefore, option (B) is the correct choice.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.