Answer :
To convert the given expression [tex]\((7x^3y^2)^{\frac{2}{7}}\)[/tex] into radical form, let's go through the process step by step:
1. Understanding the Exponents:
- The expression is [tex]\((7x^3y^2)^{\frac{2}{7}}\)[/tex].
- In terms of radicals, an expression [tex]\(a^{\frac{m}{n}}\)[/tex] can be written as [tex]\(\sqrt[n]{a^m}\)[/tex].
2. Apply the Rule:
- Here, the base [tex]\(a\)[/tex] is [tex]\(7x^3y^2\)[/tex], [tex]\(m = 2\)[/tex], and [tex]\(n = 7\)[/tex].
- So, the expression can be rewritten as [tex]\(\sqrt[7]{(7x^3y^2)^2}\)[/tex].
3. Simplifying the Power:
- [tex]\((7x^3y^2)^2\)[/tex] means every component of the inside expression is raised to the power of 2.
- Calculate: [tex]\(7^2 = 49\)[/tex], [tex]\((x^3)^2 = x^{6}\)[/tex], [tex]\((y^2)^2 = y^{4}\)[/tex].
- This simplifies to: [tex]\(49x^6y^4\)[/tex].
4. Express in Radical Form:
- Put everything under the 7th root: [tex]\(\sqrt[7]{49x^6y^4}\)[/tex].
So, the given expression [tex]\((7x^3y^2)^{\frac{2}{7}}\)[/tex] in radical form is [tex]\(\sqrt[7]{49x^6y^4}\)[/tex].
The answer is:
[tex]\(\sqrt[7]{49x^6y^4}\)[/tex]
1. Understanding the Exponents:
- The expression is [tex]\((7x^3y^2)^{\frac{2}{7}}\)[/tex].
- In terms of radicals, an expression [tex]\(a^{\frac{m}{n}}\)[/tex] can be written as [tex]\(\sqrt[n]{a^m}\)[/tex].
2. Apply the Rule:
- Here, the base [tex]\(a\)[/tex] is [tex]\(7x^3y^2\)[/tex], [tex]\(m = 2\)[/tex], and [tex]\(n = 7\)[/tex].
- So, the expression can be rewritten as [tex]\(\sqrt[7]{(7x^3y^2)^2}\)[/tex].
3. Simplifying the Power:
- [tex]\((7x^3y^2)^2\)[/tex] means every component of the inside expression is raised to the power of 2.
- Calculate: [tex]\(7^2 = 49\)[/tex], [tex]\((x^3)^2 = x^{6}\)[/tex], [tex]\((y^2)^2 = y^{4}\)[/tex].
- This simplifies to: [tex]\(49x^6y^4\)[/tex].
4. Express in Radical Form:
- Put everything under the 7th root: [tex]\(\sqrt[7]{49x^6y^4}\)[/tex].
So, the given expression [tex]\((7x^3y^2)^{\frac{2}{7}}\)[/tex] in radical form is [tex]\(\sqrt[7]{49x^6y^4}\)[/tex].
The answer is:
[tex]\(\sqrt[7]{49x^6y^4}\)[/tex]
