Write the following in simplified radical form:
$\sqrt[3]{250 y^8}$
To simplify the expression \(\sqrt[3]{250y^8}\), we begin by examining the components inside the cube root. The expression consists of a numerical coefficient \(250\) and a variable part \(y^8\). A cube root \(\sqrt[3]{x}\) is equivalent to taking the power of \(1/3\). Our objective is to factor out any perfect cubes from the radicand, as they can be extracted from the radical.
First, we factor the numerical coefficient \(250\). We look for the largest perfect cube factor:
\[250 = 125 \times 2 = 5^3 \times 2.\]
Next, we factor the variable term \(y^8\). We want to express the exponent as a multiple of 3 plus a remainder:
\[y^8 = y^6 \times y^2 = (y^2)^3 \times y^2.\]
These factorizations allow us to separate the perfect cube components from the rest of the expression.
We rewrite the original expression by substituting our factors back into the cube root:
\[\sqrt[3]{250y^8} = \sqrt[3]{5^3 \cdot 2 \cdot (y^2)^3 \cdot y^2}.\]
Using the property of radicals \(\sqrt[3]{abc} = \sqrt[3]{a} \cdot \sqrt[3]{b} \cdot \sqrt[3]{c}\), we can distribute the cube root:
\[\sqrt[3]{5^3} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{2y^2}.\]
Evaluating the perfect cubes \(\sqrt[3]{5^3} = 5\) and \(\sqrt[3]{(y^2)^3} = y^2\), we obtain:
\[5y^2 \sqrt[3]{2y^2}.\]
This is the fully simplified form of the given expression.
Simplifying radical expressions is a fundamental skill in algebra, useful for solving equations, working with geometric formulas (such as the volume of a sphere or the length of a space diagonal), and engineering designs. For example, if you need to calculate the side length of a cube with volume \(250y^8\), you would need to compute \(\sqrt[3]{250y^8}\), which simplifies to \(5y^2\sqrt[3]{2y^2}\). This makes it easier to estimate the value or use it in further calculations.