Questions: Combine. Assume that all variables represent nonnegative real numbers. 7 sqrt(32 x)-5 sqrt(18 x)

Transcript text: Combine. Assume that all variables represent nonnegative real numbers. \[ 7 \sqrt{32 x}-5 \sqrt{18 x} \]

Solution

Solution Steps

Step 1: Original Expression

We start with the expression: \[ 7 \sqrt{32 x} - 5 \sqrt{18 x} \]

Step 2: Simplifying Each Term

First, we simplify each square root term: \[ \sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} = 4 \sqrt{2} \sqrt{x} \] \[ \sqrt{18 x} = \sqrt{9 \cdot 2 \cdot x} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{x} = 3 \sqrt{2} \sqrt{x} \]

Step 3: Substitute Back into the Expression

Now we substitute these simplified terms back into the original expression: \[ 7 \cdot 4 \sqrt{2} \sqrt{x} - 5 \cdot 3 \sqrt{2} \sqrt{x} \] This simplifies to: \[ 28 \sqrt{2} \sqrt{x} - 15 \sqrt{2} \sqrt{x} \]

Step 4: Combine Like Terms

Now we combine the like terms: \[ (28 - 15) \sqrt{2} \sqrt{x} = 13 \sqrt{2} \sqrt{x} \]