- Factorize the numerical coefficient: \(1250 = 5^3 \times 10\).
- Rewrite variable powers as cubes: \(x^{12} = (x^4)^3\) and \(y^7 = (y^2)^3 \cdot y\).
- Extract the cube roots of the perfect cube components.
- Combine the results to get the final simplified expression: \(\boxed{5 x^4 y^2 \sqrt[3]{10 y}}\).
Explanation
- Analyzing the numerical part
To simplify the expression \(\sqrt[3]{1250 x^{12} y^7}\), we look for factors that are perfect cubes. A perfect cube is a number or variable raised to the power of three (or a multiple of three). We start by breaking down the numerical coefficient \(1250\) into its prime factors or identifying its largest perfect cube divisor. We note that \(1250 = 125 \times 10 = 5^3 \times 10\). Since \(125 = 5^3\), we can pull the factor of \(5\) out of the cube root.
- Simplifying the variable parts
Next, we examine the variables \(x^{12}\) and \(y^7\). We apply the rule of exponents \((a^m)^n = a^{mn}\) in reverse to write these variables as powers of three.
For \(x^{12}\), we see that \(12 = 3 \times 4\), so \(x^{12} = (x^4)^3\). This is a perfect cube, meaning \(\sqrt[3]{x^{12}} = x^4\).
For \(y^7\), we can write this as \(y^6 \times y = (y^2)^3 \times y\). Here, \((y^2)^3\) is a perfect cube, while the remaining factor \(y\) must stay inside the cube root.
- Combining and simplifying the full expression
Now we combine everything back together under the cube root:
\[\sqrt[3]{1250 x^{12} y^7} = \sqrt[3]{(5^3 \cdot 10) \cdot (x^4)^3 \cdot (y^2)^3 \cdot y}.\]
Using the property of radicals \(\sqrt[n]{abc} = \sqrt[n]{a} \cdot \sqrt[n]{b} \cdot \sqrt[n]{c}\), we can separate the perfect cube factors from the non-cube factors:
\[\sqrt[3]{5^3} \cdot \sqrt[3]{(x^4)^3} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{10 \cdot y} = 5 \cdot x^4 \cdot y^2 \cdot \sqrt[3]{10 y}.\]
Simplifying this, we get \(5 x^4 y^2 \sqrt[3]{10 y}\).
- Final result and comparison
Comparing our result, \(5 x^4 y^2 \sqrt[3]{10 y}\), with the provided options, we see it matches the third option exactly. Therefore, the simplified expression is \(\boxed{5 x^4 y^2 \sqrt[3]{10 y}}\).
Examples
In fields like structural engineering or physics, engineers often deal with volume calculations where dimensions are expressed as variables. When calculating the cube root of a complex volume expression, simplifying the radical allows for much easier comparisons and standardizes results. For instance, if a mechanical component has a volume represented by a complex cube root expression, simplifying it helps in determining the necessary material thickness or cross-sectional area quickly and accurately.