If \(x=y^2, y=z^2\) and \((x y z)^4 / 2=64, x\) is: (A) 2 (B) 4 (C) 8 (D) 1 (E) 16

In this exercise, variable substitution is key. Given the equations: \ \(x = y^2\), \ \(y = z^2\), \ and \ \((xyz)^4 / 2 = 64\), we start by expressing everything in terms of one variable. This makes solving easier and more systematic. For example, using \(y = z^2\), you can substitute \(z\) into the equations for \(x\) and \(y\). This gives you: \ \(x = (z^2)^2\) which simplifies to \(x = z^4\). Now, both \(x\) and \(y\) are expressed in terms of \(z\), making subsequent calculations straightforward.

Solving the given algebraic equations involves combining and simplifying them. After substitution, the third equation becomes: \ \((z^4 \times z^2 \times z)^4 / 2 = 64\). \ Simplify inside the parentheses by combining the exponents: \ \((z^4 \times z^2 \times z) = z^{4+2+1} = z^7\). \ This reduces the equation to: \ \((z^7)^4 / 2 = 64\). By raising \(z^7\) to the power of 4, you get \(z^{28}\), so the equation now is: \ \(z^{28} / 2 = 64\). \ Multiplying both sides by 2, you solve for \(z^{28}\), which gives: \ \(z^{28} = 128\). \ Taking the 28th root of both sides: \ \(z = 128^{1/28}\). Finally, knowing \(128 = 2^7\), this simplifies to \(z = 2^{1/4}\).

Let's break down the solution steps: \ 1. **Understand the given equations**: Identify relationships \(x = y^2\), \(y = z^2\), and \((xyz)^4 / 2 = 64\). \ 2. **Substitute values**: Use \(y = z^2\) to express \(x\) as \(x = z^4\). Now all variables are in terms of \(z\). \ 3. **Combine into a single equation**: Substitute back into \((xyz)^4 / 2 = 64\). \ 4. **Simplify the equation**: Simplify the variables inside the parentheses and solve for \(z\). \ 5. **Solve for \(z\)**: Isolate \(z\) by taking roots. \ 6. **Find \(x\)**: Substitute \(z\) back into \(x = z^4\) and simplify to find \(x\). In this case, we solved \(128^{1/28}\) to get \(z = 2^{1/4}\) and then found \(x = (2^{-2})^4 = 2^{-8}\). \ Each of these steps builds upon the previous, making complex problems more manageable with organized thinking and methodical problem-solving.