Answer :
To find the exponential equivalent of [tex]\(\sqrt[7]{x^9}\)[/tex], we need to use the properties of exponents and roots.
1. Understanding the Expression:
- The expression [tex]\(\sqrt[7]{x^9}\)[/tex] is asking for the 7th root of [tex]\(x^9\)[/tex].
2. Rewriting Using Exponent Rules:
- A root can be expressed as a fractional exponent. Specifically, the [tex]\(n\)[/tex]th root of a number is the same as raising that number to the power of [tex]\(\frac{1}{n}\)[/tex].
- So, [tex]\(\sqrt[7]{x^9}\)[/tex] can be written as [tex]\((x^9)^{\frac{1}{7}}\)[/tex].
3. Applying the Power of a Power Rule:
- According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- Here, we have [tex]\((x^9)^{\frac{1}{7}}\)[/tex], which can be rewritten as [tex]\(x^{9 \cdot \frac{1}{7}}\)[/tex].
4. Calculating the Exponent:
- Multiply the exponents together: [tex]\(9 \times \frac{1}{7} = \frac{9}{7}\)[/tex].
- Therefore, [tex]\((x^9)^{\frac{1}{7}} = x^{\frac{9}{7}}\)[/tex].
5. Final Result:
- The exponential equivalent of [tex]\(\sqrt[7]{x^9}\)[/tex] is [tex]\(x^{\frac{9}{7}}\)[/tex].
Thus, the correct option is:
c. [tex]\(x^{\frac{9}{7}}\)[/tex]
1. Understanding the Expression:
- The expression [tex]\(\sqrt[7]{x^9}\)[/tex] is asking for the 7th root of [tex]\(x^9\)[/tex].
2. Rewriting Using Exponent Rules:
- A root can be expressed as a fractional exponent. Specifically, the [tex]\(n\)[/tex]th root of a number is the same as raising that number to the power of [tex]\(\frac{1}{n}\)[/tex].
- So, [tex]\(\sqrt[7]{x^9}\)[/tex] can be written as [tex]\((x^9)^{\frac{1}{7}}\)[/tex].
3. Applying the Power of a Power Rule:
- According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- Here, we have [tex]\((x^9)^{\frac{1}{7}}\)[/tex], which can be rewritten as [tex]\(x^{9 \cdot \frac{1}{7}}\)[/tex].
4. Calculating the Exponent:
- Multiply the exponents together: [tex]\(9 \times \frac{1}{7} = \frac{9}{7}\)[/tex].
- Therefore, [tex]\((x^9)^{\frac{1}{7}} = x^{\frac{9}{7}}\)[/tex].
5. Final Result:
- The exponential equivalent of [tex]\(\sqrt[7]{x^9}\)[/tex] is [tex]\(x^{\frac{9}{7}}\)[/tex].
Thus, the correct option is:
c. [tex]\(x^{\frac{9}{7}}\)[/tex]
