## Step : Simplify Numerator numerator is \(sqrt{y4} \- Recall that \sqrt{x^} = || \), so \( \sqrt{y^4} = |y^2| \). - Since \( y^2 \) is always non-negative for all real \( y \), \( |y^2| = y^2 \). **Explanation:** Taking the square root of \( y^4 \) gives \( |y^2| \), and since squaring any real number makes it non-negative, the absolute value is not needed. --- ## Step 2: Simplify the Denominator The denominator is \( y \). So the expression becomes: \[ \frac{y^2}{y} \] Now, simplify by dividing \( y^2 \) by \( y \): \[ \frac{y^2}{y} = y \quad \text{(for } y \neq \text{)} \] **Explanation:** Dividing \( y^2 \) by \( y \) follows the exponent rule \( a^m / a^n = a^{m-n} \). Here, \( y^{2-1} = y^1 = y \). --- ## Final Answer \[ \frac{\sqrt{y^4}}{y} = y \quad \text{(for } y \neq \text{)} \]

## Step 1: Simplify the Numerator The numerator is \(\sqrt{y^4}\). Recall that \(\sqrt{x^2} = |x|\), so: \[ \sqrt{y^4} = |y^2| \] Since \(y^2\) is always non-negative for all real \(y\), we can conclude that: \[ |y^2| = y^2 \] **Explanation:** Taking the square root of \(y^4\) results in \(|y^2|\), but since \(y^2\) is non-negative, we can drop the absolute value to get \(y^2\). --- ## Step 2: Simplify the Denominator The denominator is \(y\). Thus, we can rewrite the expression: \[ \frac{y^2}{y} \] Now, simplify by dividing \(y^2\) by \(y\): \[ \frac{y^2}{y} = y \quad \text{(for } y \neq 0\text{)} \] **Explanation:** Dividing \(y^2\) by \(y\) utilizes the exponent rule \(a^m / a^n = a^{m-n}\). Specifically, we have \(y^{2-1} = y^1 = y\). --- ## Final Answer \[ \frac{\sqrt{y^4}}{y} = y \quad \text{(for } y \neq 0\text{)} \]