Question

which two choices are equivalent to this expression?
$2\sqrt{75} + 3\sqrt{50}$
a $2\sqrt{25\cdot3} + 3\sqrt{25\cdot2}$
b $3\sqrt{25} + 2\sqrt{25}$
c $10\sqrt{3} + 15\sqrt{2}$
d $25\sqrt{3} + 25\sqrt{2}$
e $25\sqrt{6}$

Explanation:

Step1: Simplify \(2\sqrt{75}\)

Factor 75: \(75 = 25\times3\), so \(\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}\). Then \(2\sqrt{75}=2\times5\sqrt{3}=10\sqrt{3}\).

Step2: Simplify \(3\sqrt{50}\)

Factor 50: \(50 = 25\times2\), so \(\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}\). Then \(3\sqrt{50}=3\times5\sqrt{2}=15\sqrt{2}\).

Step3: Combine the simplified terms

The original expression \(2\sqrt{75}+3\sqrt{50}\) becomes \(10\sqrt{3}+15\sqrt{2}\) (from Step1 and Step2). Now check the options:

• Option A: \(2\sqrt{25\cdot3}+3\sqrt{25\cdot2}=2\times5\sqrt{3}+3\times5\sqrt{2}=10\sqrt{3}+15\sqrt{2}\) (matches after simplifying, same as original simplified).
• Option B: \(3\sqrt{25}+2\sqrt{25}=3\times5 + 2\times5=15 + 10 = 25\) (does not match).
• Option C: \(10\sqrt{3}+15\sqrt{2}\) (matches the simplified original).
• Option D: \(25\sqrt{3}+25\sqrt{2}\) (does not match).
• Option E: \(25\sqrt{6}\) (does not match).

Wait, but the problem says "which two choices" – let's re - check Option A's simplification. \(2\sqrt{25\cdot3}=2\times5\sqrt{3}=10\sqrt{3}\), \(3\sqrt{25\cdot2}=3\times5\sqrt{2}=15\sqrt{2}\), so Option A is \(10\sqrt{3}+15\sqrt{2}\) (same as Option C and the original simplified). Wait, maybe a typo in Option A's display? Wait, the original Option A is written as \(2\sqrt{25\cdot3}+3\sqrt{25\cdot2}\), which simplifies to \(10\sqrt{3}+15\sqrt{2}\), and Option C is \(10\sqrt{3}+15\sqrt{2}\). Wait, maybe the user's image has Option A as \(2\sqrt{25\times3}+3\sqrt{25\times2}\), which is correct. So the two equivalent choices are A and C? Wait, no, let's re - calculate:

Wait, \(2\sqrt{75}=2\sqrt{25\times3}=2\times5\sqrt{3}=10\sqrt{3}\), \(3\sqrt{50}=3\sqrt{25\times2}=3\times5\sqrt{2}=15\sqrt{2}\), so \(2\sqrt{75}+3\sqrt{50}=10\sqrt{3}+15\sqrt{2}\).

Option A: \(2\sqrt{25\times3}+3\sqrt{25\times2}=2\times5\sqrt{3}+3\times5\sqrt{2}=10\sqrt{3}+15\sqrt{2}\) (correct).

Option C: \(10\sqrt{3}+15\sqrt{2}\) (correct).

Wait, but in the image, Option A is written as \(2\sqrt{25\cdot3}+3\sqrt{25\cdot2}\), which is the factored form before multiplying out the 5s, but when simplified, it's the same as Option C. So the two choices are A and C.

Answer:

A. \(2\sqrt{25\cdot3}+3\sqrt{25\cdot2}\), C. \(10\sqrt{3}+15\sqrt{2}\)