Answer :
To determine which choice is equivalent to the product [tex]\(\sqrt{5 x} \cdot \sqrt{x+3}\)[/tex], we can use properties of square roots and algebraic manipulation.
### Step-by-step solution:
1. Use the property of square roots:
The property of square roots that we will use is:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property to the given expression [tex]\(\sqrt{5 x} \cdot \sqrt{x + 3}\)[/tex], we get:
[tex]\[ \sqrt{5 x} \cdot \sqrt{x+3} = \sqrt{(5 x) \cdot (x + 3)} \][/tex]
2. Simplify the expression within the square root:
Next, we simplify the product inside the square root:
[tex]\[ (5 x) \cdot (x + 3) = 5 x \cdot x + 5 x \cdot 3 \][/tex]
Simplifying further, we get:
[tex]\[ 5 x^2 + 15 x \][/tex]
3. Combine the constants and variables:
Our simplified expression inside the square root is:
[tex]\[ 5 x^2 + 15 x \][/tex]
4. Write the equivalent expression:
Now that we have simplified the product inside the square root, we can write the equivalent expression as:
[tex]\[ \sqrt{5 x^2 + 15 x} \][/tex]
### Conclusion:
Thus, the expression [tex]\(\sqrt{5 x} \cdot \sqrt{x + 3}\)[/tex] simplifies to [tex]\(\sqrt{5 x^2 + 15 x}\)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{\sqrt{5 x^2 + 15 x}} \][/tex]
So, the correct answer is option C: [tex]\(\sqrt{5 x^2 + 15 x}\)[/tex].
### Step-by-step solution:
1. Use the property of square roots:
The property of square roots that we will use is:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property to the given expression [tex]\(\sqrt{5 x} \cdot \sqrt{x + 3}\)[/tex], we get:
[tex]\[ \sqrt{5 x} \cdot \sqrt{x+3} = \sqrt{(5 x) \cdot (x + 3)} \][/tex]
2. Simplify the expression within the square root:
Next, we simplify the product inside the square root:
[tex]\[ (5 x) \cdot (x + 3) = 5 x \cdot x + 5 x \cdot 3 \][/tex]
Simplifying further, we get:
[tex]\[ 5 x^2 + 15 x \][/tex]
3. Combine the constants and variables:
Our simplified expression inside the square root is:
[tex]\[ 5 x^2 + 15 x \][/tex]
4. Write the equivalent expression:
Now that we have simplified the product inside the square root, we can write the equivalent expression as:
[tex]\[ \sqrt{5 x^2 + 15 x} \][/tex]
### Conclusion:
Thus, the expression [tex]\(\sqrt{5 x} \cdot \sqrt{x + 3}\)[/tex] simplifies to [tex]\(\sqrt{5 x^2 + 15 x}\)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{\sqrt{5 x^2 + 15 x}} \][/tex]
So, the correct answer is option C: [tex]\(\sqrt{5 x^2 + 15 x}\)[/tex].
