Simplify 4$\sqrt{27}$ + 5$\sqrt{12}$ − 3$\sqrt{75}$

Analysis

To simplify this expression, we need to express each square root in its simplest surd form. The goal is to break down the numbers inside the square roots (the radicands) into factors where at least one factor is a perfect square.

Concept: Simplifying Surds

A surd like $\sqrt{n}$ can be simplified if $n$ has a factor that is a perfect square (e.g., 4, 9, 16, 25, 36...). We use the rule:

By identifying a perfect square factor, we can extract its square root and leave the remaining prime factor under the radical.

Step-by-Step Solution

Step 1: Simplify each term individually.

• For $4\sqrt{27}$: $27 = 9 \times 3$ $4\sqrt{27} = 4\sqrt{9 \times 3} = 4 \times 3\sqrt{3} = 12\sqrt{3}$

• For $5\sqrt{12}$: $12 = 4 \times 3$ $5\sqrt{12} = 5\sqrt{4 \times 3} = 5 \times 2\sqrt{3} = 10\sqrt{3}$

• For $3\sqrt{75}$: $75 = 25 \times 3$ $3\sqrt{75} = 3\sqrt{25 \times 3} = 3 \times 5\sqrt{3} = 15\sqrt{3}$

Step 2: Substitute the simplified terms back into the original expression. The expression becomes:

Step 3: Combine the like terms. Since all terms now have $\sqrt{3}$, we simply add and subtract the coefficients:

Conclusion

Comparing our result with the given options, we find that $7\sqrt{3}$ matches option D.

Exam Tip

When you see multiple surds in an addition or subtraction problem, they almost always share a common "root" (in this case, $\sqrt{3}$). If you get stuck, try dividing the numbers inside the roots by small prime numbers like 2, 3, or 5 to see which one leaves you with a perfect square!