Simplifying Radical Expressions: √12 + √75

The goal is to simplify the expression $\sqrt{12} + \sqrt{75}$ by simplifying each square root term first.

Step 1: Simplify √12

Find the largest perfect square that divides 12. The largest perfect square factor of 12 is 4 ($2^2 = 4$).

• Rewrite $\sqrt{12}$ using the factor: $\sqrt{12} = \sqrt{4 \times 3}$
• Separate the square roots: $\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3}$
• Simplify: $\sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

So, $\sqrt{12} = 2\sqrt{3}$.

Step 2: Simplify √75

Find the largest perfect square that divides 75. The largest perfect square factor of 75 is 25 ($5^2 = 25$).

• Rewrite $\sqrt{75}$ using the factor: $\sqrt{75} = \sqrt{25 \times 3}$
• Separate the square roots: $\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}$
• Simplify: $\sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

So, $\sqrt{75} = 5\sqrt{3}$.

Step 3: Add the Simplified Radicals

Now substitute the simplified terms back into the original expression:

$\sqrt{12} + \sqrt{75} = 2\sqrt{3} + 5\sqrt{3}$

Step 4: Combine Like Terms

Since both terms contain $\sqrt{3}$, they are like terms. Add the coefficients (the numbers in front of the radical):

$(2 + 5)\sqrt{3} = 7\sqrt{3}$

Final Answer

The simplified form of $\sqrt{12} + \sqrt{75}$ is $7\sqrt{3}$.