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Which choice is equivalent to the product below for acceptable values of [tex]\( x \)[/tex]?

[tex]\[ \sqrt{5x} \cdot \sqrt{x+3} \][/tex]

A. [tex]\( \sqrt{5x^2 + 15} \)[/tex]
B. [tex]\( \sqrt{5x^2 + 3} \)[/tex]
C. [tex]\( \sqrt{5x^2 + 15x} \)[/tex]
D. [tex]\( 5x \sqrt{x+3} \)[/tex]


Sagot :

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].