Simplify ninth root of 125x^3y^3

Solution:

Step 1:

Express $125x^3y^3$ as $(5xy)^3$. Therefore, we have $\sqrt[9]{(5xy)^3}$.

Step 2:

Transform $\sqrt[9]{(5xy)^3}$ into the compound radical $\sqrt[3]{\sqrt[3]{(5xy)^3}}$.

Step 3:

Extract the terms from under the radical, given that all numbers are real. The result is $\sqrt[3]{5xy}$.

Knowledge Notes:

To simplify the ninth root of $125x^3y^3$, we follow these steps:

1. Rewriting the Expression: We start by recognizing that $125$ is a perfect cube, as $125 = 5^3$. Since we also have $x^3$ and $y^3$, we can rewrite the entire expression as $(5xy)^3$.

2. Understanding Radicals: The ninth root of a number can be expressed as a radical with an index of 9. In this case, we have $\sqrt[9]{(5xy)^3}$. This can be further simplified by realizing that taking the ninth root is the same as raising to the power of $\frac{1}{9}$, and we can use the property of radicals that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.

3. Simplifying Nested Radicals: We can simplify the expression by considering the ninth root of a cube as a third root of a third root, which is written as $\sqrt[3]{\sqrt[3]{(5xy)^3}}$. This is because taking the third root twice is equivalent to taking the ninth root (since $3 \times 3 = 9$).

4. Extraction of Terms: Finally, we use the property that the cube root of a cube is the number itself, so $\sqrt[3]{(5xy)^3} = 5xy$. Since we're taking the cube root of a cube root, we end up with just $\sqrt[3]{5xy}$.

Relevant properties of radicals and exponents used in this process include:

• $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

• $(a^m)^n = a^{mn}$

• If $a$ is a non-negative real number and $m$ is an integer, then $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ provided that the result is a real number.

• Nested radicals can be simplified by multiplying the indices if the radicand remains the same.

Understanding these properties is crucial for simplifying expressions involving roots and exponents.