Problem:
Simplify the following expression:
- 3√32 + 4√50 − 5√72 + √98
Solution (Step-by-Step):
Step 1: Factorize the radicands into their prime factors.
We aim to find perfect squares within each radicand to simplify the square roots:
- √32 = √16 × √2 = 4√2
- √50 = √25 × √2 = 5√2
- √72 = √36 × √2 = 6√2
- √98 = √49 × √2 = 7√2
Step 2: Substitute the simplified square roots back into the expression
The expression becomes:
- 3√32 + 4√50 − 5√72 + √98 = 3(4√2) + 4(5√2) − 5(6√2) + 7√2
Simplify the coefficients:
- 12√2 + 20√2 − 30√2 + 7√2
Step 3: Combine like terms
Add and subtract the coefficients of √2:
- 12 + 20 − 30 + 7 = 9
Thus, the simplified expression is:
- 9√2
Final Answer:
- 9√2
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