Combine the following expressions. \( a \sqrt{125 y}-b \sqrt{45 y} \) \( (5 a+3 b) \sqrt{5 y} \) \( (5 a-3 b) \sqrt{5 y} \) \( (-5 a-3 b) \sqrt{5 y} \)

Simplify the expression by following steps: - step0: Distribute \(\sqrt{5y}\): \(\left(-5a-3b\right)\sqrt{5y}\) - step1: Distribute \(\sqrt{5y}:\) \(-5a\sqrt{5y}-3b\sqrt{5y}\) Expand the expression \( (5 a+3 b) \sqrt{5 y} \) Simplify the expression by following steps: - step0: Distribute \(\sqrt{5y}\): \(\left(5a+3b\right)\sqrt{5y}\) - step1: Distribute \(\sqrt{5y}:\) \(5a\sqrt{5y}+3b\sqrt{5y}\) Expand the expression \( a \sqrt{125 y}-b \sqrt{45 y} \) Simplify the expression by following steps: - step0: Simplify: \(a\sqrt{125y}-b\sqrt{45y}\) - step1: Simplify the root: \(a\times 5\sqrt{5y}-b\sqrt{45y}\) - step2: Simplify the root: \(a\times 5\sqrt{5y}-b\times 3\sqrt{5y}\) - step3: Reorder the terms: \(5a\sqrt{5y}-b\times 3\sqrt{5y}\) - step4: Reorder the terms: \(5a\sqrt{5y}-3b\sqrt{5y}\) Expand the expression \( (5 a-3 b) \sqrt{5 y} \) Simplify the expression by following steps: - step0: Distribute \(\sqrt{5y}\): \(\left(5a-3b\right)\sqrt{5y}\) - step1: Distribute \(\sqrt{5y}:\) \(5a\sqrt{5y}-3b\sqrt{5y}\) Let's combine the given expressions step by step. 1. **First Expression:** \[ a \sqrt{125 y} - b \sqrt{45 y} \] We can simplify this: \[ \sqrt{125 y} = \sqrt{25 \cdot 5 \cdot y} = 5 \sqrt{5y} \] \[ \sqrt{45 y} = \sqrt{9 \cdot 5 \cdot y} = 3 \sqrt{5y} \] Therefore, the first expression simplifies to: \[ 5a \sqrt{5y} - 3b \sqrt{5y} \] 2. **Second Expression:** \[ (5a + 3b) \sqrt{5y} \] This expression is already in a simplified form. 3. **Third Expression:** \[ (5a - 3b) \sqrt{5y} \] This expression is also in a simplified form. 4. **Fourth Expression:** \[ (-5a - 3b) \sqrt{5y} \] This expression is also in a simplified form. Now, let's combine all the simplified expressions: - From the first expression: \( 5a \sqrt{5y} - 3b \sqrt{5y} \) - From the second expression: \( 5a \sqrt{5y} + 3b \sqrt{5y} \) - From the third expression: \( 5a \sqrt{5y} - 3b \sqrt{5y} \) - From the fourth expression: \( -5a \sqrt{5y} - 3b \sqrt{5y} \) Now, we can combine these: \[ (5a \sqrt{5y} - 3b \sqrt{5y}) + (5a \sqrt{5y} + 3b \sqrt{5y}) + (5a \sqrt{5y} - 3b \sqrt{5y}) + (-5a \sqrt{5y} - 3b \sqrt{5y}) \] Combining like terms: - The \(5a \sqrt{5y}\) terms: \(5a + 5a + 5a - 5a = 10a\) - The \(-3b \sqrt{5y}\) and \(+3b \sqrt{5y}\) terms cancel out. - The \(-3b \sqrt{5y}\) from the first and third expressions and \(-3b \sqrt{5y}\) from the fourth expression gives us: \(-3b - 3b = -6b\) Thus, the final combined expression is: \[ (10a - 6b) \sqrt{5y} \]