- Factor the constants inside the radicals to simplify the left-hand sides: \(\sqrt{448x^c} = 8\sqrt{7}x^{c/2}\) and \(\sqrt[3]{576x^d} = 4\sqrt[3]{9}x^{d/3}\).
- Express the right-hand sides using rational exponents: \(8x^3 \sqrt{7x} = 8\sqrt{7}x^{7/2}\) and \(4x \sqrt[3]{9x^2} = 4\sqrt[3]{9}x^{5/3}\).
- Compare the exponents to find the values of \(c\) and \(d\).
- The values are \(c = \boxed{7}\) and \(d = \boxed{5}\).
Explanation
- Problem Analysis and Plan
We are given two equations for \(x > 0\) and need to determine the values of constants \(c\) and \(d\).
Equation A: \(\sqrt{448x^c} = 8x^3 \sqrt{7x}\)
Equation B: \(\sqrt[3]{576x^d} = 4x \sqrt[3]{9x^2}\)
Our plan is to simplify both sides of each equation using properties of radicals and exponents, then equate the exponents of \(x\) to find the unknown constants. Recall that \(\sqrt[n]{x^m} = x^{m/n}\) and \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\).
- Solving for c in Equation A
For Equation A, we start by simplifying the left-hand side:
\(\sqrt{448x^c} = \sqrt{64 \cdot 7 \cdot x^c} = \sqrt{64} \cdot \sqrt{7} \cdot \sqrt{x^c} = 8\sqrt{7}x^{c/2}\)
Now, we simplify the right-hand side:
\(8x^3 \sqrt{7x} = 8 \cdot x^3 \cdot \sqrt{7} \cdot \sqrt{x} = 8\sqrt{7} \cdot x^3 \cdot x^{1/2} = 8\sqrt{7} \cdot x^{3 + 1/2} = 8\sqrt{7} \cdot x^{7/2}\)
By equating the two sides:
\(8\sqrt{7}x^{c/2} = 8\sqrt{7}x^{7/2}\)
This implies \(\frac{c}{2} = \frac{7}{2}\), which gives \(c = 7\).
- Solving for d in Equation B
For Equation B, we simplify the left-hand side:
\(\sqrt[3]{576x^d} = \sqrt[3]{64 \cdot 9 \cdot x^d} = \sqrt[3]{64} \cdot \sqrt[3]{9} \cdot \sqrt[3]{x^d} = 4\sqrt[3]{9} \cdot x^{d/3}\)
Next, we simplify the right-hand side:
\(4x \sqrt[3]{9x^2} = 4 \cdot x \cdot \sqrt[3]{9} \cdot \sqrt[3]{x^2} = 4\sqrt[3]{9} \cdot x^1 \cdot x^{2/3} = 4\sqrt[3]{9} \cdot x^{1 + 2/3} = 4\sqrt[3]{9} \cdot x^{5/3}\)
By equating the two sides:
\(4\sqrt[3]{9} \cdot x^{d/3} = 4\sqrt[3]{9} \cdot x^{5/3}\)
This implies \(\frac{d}{3} = \frac{5}{3}\), which gives \(d = 5\).
- Conclusion
We have determined that \(c = 7\) and \(d = 5\). These values satisfy the given radical equations. Thus, in equation A, \(c = \boxed{7}\), and in equation B, \(d = \boxed{5}\).
Examples
In many engineering and physics applications, dimensions and units of measurement are analyzed using exponents and radicals to ensure consistency. For example, when calculating the velocity of a fluid in a pipe where the diameter relates to the flow rate through a square or cube root relationship, understanding how to manipulate these expressions is crucial. By isolating variables like \(c\) and \(d\), we can ensure that our mathematical models correctly represent the physical phenomenon, such as scaling a structural design based on force or pressure variables.