Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Certainly! Let's solve the expression [tex]\( 9^{\frac{2}{3}} \)[/tex] and simplify it to find out which value remains under the radical.
1. Express [tex]\( 9 \)[/tex] in terms of its prime factors:
The number [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex]. Thus, the expression [tex]\( 9^{\frac{2}{3}} \)[/tex] can be rewritten using its prime factors:
[tex]\[ 9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}} \][/tex]
2. Simplify the exponent:
Using the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify [tex]\((3^2)^{\frac{2}{3}}\)[/tex]:
[tex]\[ (3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}} \][/tex]
3. Multiply the exponents:
Next, we multiply [tex]\( 2 \)[/tex] and [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ 3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}} \][/tex]
4. Express in simplest radical form:
The exponent [tex]\( \frac{4}{3} \)[/tex] indicates that the base [tex]\( 3 \)[/tex] is raised to the fourth power and then the cube root is taken. This can be written as:
[tex]\[ 3^{\frac{4}{3}} = \sqrt[3]{3^4} \][/tex]
Since [tex]\( 3^4 = 81 \)[/tex], we have:
[tex]\[ \sqrt[3]{3^4} = \sqrt[3]{81} \][/tex]
So, in the given notation [tex]\( 9^{\frac{2}{3}} \)[/tex], when simplified to its radical form, we get [tex]\( \sqrt[3]{81} \)[/tex].
The value that remains under the radical is the base value of [tex]\( 3 \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{3} \][/tex]
1. Express [tex]\( 9 \)[/tex] in terms of its prime factors:
The number [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex]. Thus, the expression [tex]\( 9^{\frac{2}{3}} \)[/tex] can be rewritten using its prime factors:
[tex]\[ 9^{\frac{2}{3}} = (3^2)^{\frac{2}{3}} \][/tex]
2. Simplify the exponent:
Using the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify [tex]\((3^2)^{\frac{2}{3}}\)[/tex]:
[tex]\[ (3^2)^{\frac{2}{3}} = 3^{2 \cdot \frac{2}{3}} \][/tex]
3. Multiply the exponents:
Next, we multiply [tex]\( 2 \)[/tex] and [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ 3^{2 \cdot \frac{2}{3}} = 3^{\frac{4}{3}} \][/tex]
4. Express in simplest radical form:
The exponent [tex]\( \frac{4}{3} \)[/tex] indicates that the base [tex]\( 3 \)[/tex] is raised to the fourth power and then the cube root is taken. This can be written as:
[tex]\[ 3^{\frac{4}{3}} = \sqrt[3]{3^4} \][/tex]
Since [tex]\( 3^4 = 81 \)[/tex], we have:
[tex]\[ \sqrt[3]{3^4} = \sqrt[3]{81} \][/tex]
So, in the given notation [tex]\( 9^{\frac{2}{3}} \)[/tex], when simplified to its radical form, we get [tex]\( \sqrt[3]{81} \)[/tex].
The value that remains under the radical is the base value of [tex]\( 3 \)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{3} \][/tex]
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.