Question

Use numerals Instead of words. If x > 0, what values of c and d make the equations true?

Equation A :
`\sqrt{448 {x}^(c) } = 8 {x}^(3) √(7x)`
Equation B :
`\sqrt[3]{576x {}^(d) } = 4x \sqrt[3]{9x {}^(2) }`In equation A, c is ...In equation B, d is ________.

Answer 1

In Equation A, the value of c that makes the equation true is 6. In Equation B, the value of d that satisfies the equation is 5.

Step-by-step explanation:

In Equation A, we start off with \(\sqrt{448x^c} = 8x^3\sqrt{7x}\). First, we simplify the left side by factoring out squares: \(\sqrt{64 \cdot 7 \cdot x^c} = 8x^3\sqrt{7x}\) which becomes \(8\sqrt{7x^c} = 8x^3\sqrt{7x}\). Cancel out 8 and \(\sqrt{7x}\) from both sides, leaving us with \(x^{c/2} = x^3\). Therefore, \(c/2 = 3\) which implies \(c = 6\).

For Equation B, \(\sqrt[3]{576x^d} = 4x\sqrt[3]{9x^2}\). Again, we begin by simplifying: \(\sqrt[3]{64 \cdot 9 \cdot x^d} = 4x\sqrt[3]{9x^2}\) which becomes \(4\sqrt[3]{9x^d} = 4x\sqrt[3]{9x^2}\). Dividing both sides by 4 and \(\sqrt[3]{9}\), we get \(x^{d/3} = x\sqrt[3]{x^2}\), which simplifies to \(x^{d/3} = x^{1 + 2/3}\). So, \(d/3 = 1 + 2/3\), giving us \(d = 3(1 + 2/3) = 3 + 2 = 5\).