Questions: An expression is shown below. 5 sqrt(27 x) Which value of x makes the expression equivalent to 75 sqrt(3) ? A. 3 B. 5 C. 20 D. 25

Transcript text: 2. An expression is shown below. \[ 5 \sqrt{27 x} \] Which value of $x$ makes the expression equivalent to $75 \sqrt{3}$ ? A. 3 B. 5 C. 20 D. 25

Solution

Solution Steps

Step 1: Simplify the given expression

Start by simplifying the expression $5 \sqrt{27x}$. Break down $27$ into its prime factors: \[ 27 = 3^3 \] So, the expression becomes: \[ 5 \sqrt{27x} = 5 \sqrt{3^3 \cdot x} \]

Step 2: Simplify the square root

Simplify the square root by taking out the square factors: \[ \sqrt{3^3 \cdot x} = \sqrt{3^2 \cdot 3 \cdot x} = 3 \sqrt{3x} \] Now, the expression becomes: \[ 5 \cdot 3 \sqrt{3x} = 15 \sqrt{3x} \]

Step 3: Set the expression equal to the target value

Set the simplified expression equal to $75 \sqrt{3}$: \[ 15 \sqrt{3x} = 75 \sqrt{3} \]

Step 4: Solve for $x$

Divide both sides by $15$: \[ \sqrt{3x} = 5 \sqrt{3} \] Square both sides to eliminate the square root: \[ 3x = (5 \sqrt{3})^2 \] Calculate the right-hand side: \[ 3x = 25 \cdot 3 \] Simplify: \[ 3x = 75 \] Divide both sides by $3$: \[ x = 25 \]

Step 5: Verify the answer

The value of $x$ that satisfies the equation is $25$, which corresponds to option D.