Answer :
To express [tex]\(3125 x^{\frac{2}{5}}\)[/tex] in its simplest radical form, follow these steps:
1. Identify the base and exponent:
- The expression given is [tex]\(3125 \cdot x^{\frac{2}{5}}\)[/tex].
2. Simplify the numerical part:
- First, we look at the number 3125. Notice that 3125 can be written as a power of 5: [tex]\(3125 = 5^5\)[/tex].
3. Apply the properties of exponents:
- We know that [tex]\(a^{\frac{m}{n}}\)[/tex] means the [tex]\(n\)[/tex]th root of [tex]\(a\)[/tex] raised to the [tex]\(m\)[/tex]th power. So, [tex]\(3125^{\frac{2}{5}} = (5^5)^{\frac{2}{5}}\)[/tex].
- Using the property [tex]\((a^b)^c = a^{bc}\)[/tex], we get: [tex]\((5^5)^{\frac{2}{5}} = 5^{5 \cdot \frac{2}{5}} = 5^2\)[/tex].
4. Calculate the power:
- [tex]\(5^2 = 25\)[/tex].
5. Combine with the variable [tex]\(x\)[/tex]:
- For the variable part [tex]\(x^{\frac{2}{5}}\)[/tex], it remains as is because we are asked for the radical form.
- This can be expressed as [tex]\(\sqrt[5]{x^2}\)[/tex], which means the fifth root of [tex]\(x\)[/tex] squared.
6. Write the final simplified form:
- Combine both parts to get the simplest radical form: [tex]\(25 \cdot \sqrt[5]{x^2}\)[/tex].
So, the expression [tex]\(3125 x^{\frac{2}{5}}\)[/tex] simplifies to [tex]\(25 \cdot \sqrt[5]{x^2}\)[/tex] in radical form.
1. Identify the base and exponent:
- The expression given is [tex]\(3125 \cdot x^{\frac{2}{5}}\)[/tex].
2. Simplify the numerical part:
- First, we look at the number 3125. Notice that 3125 can be written as a power of 5: [tex]\(3125 = 5^5\)[/tex].
3. Apply the properties of exponents:
- We know that [tex]\(a^{\frac{m}{n}}\)[/tex] means the [tex]\(n\)[/tex]th root of [tex]\(a\)[/tex] raised to the [tex]\(m\)[/tex]th power. So, [tex]\(3125^{\frac{2}{5}} = (5^5)^{\frac{2}{5}}\)[/tex].
- Using the property [tex]\((a^b)^c = a^{bc}\)[/tex], we get: [tex]\((5^5)^{\frac{2}{5}} = 5^{5 \cdot \frac{2}{5}} = 5^2\)[/tex].
4. Calculate the power:
- [tex]\(5^2 = 25\)[/tex].
5. Combine with the variable [tex]\(x\)[/tex]:
- For the variable part [tex]\(x^{\frac{2}{5}}\)[/tex], it remains as is because we are asked for the radical form.
- This can be expressed as [tex]\(\sqrt[5]{x^2}\)[/tex], which means the fifth root of [tex]\(x\)[/tex] squared.
6. Write the final simplified form:
- Combine both parts to get the simplest radical form: [tex]\(25 \cdot \sqrt[5]{x^2}\)[/tex].
So, the expression [tex]\(3125 x^{\frac{2}{5}}\)[/tex] simplifies to [tex]\(25 \cdot \sqrt[5]{x^2}\)[/tex] in radical form.
