Simplify √72 - √32 + √48
Understand the Problem
The question is asking us to simplify the expression involving square roots: √72 - √32 + √48. We will simplify each square root separately and then combine the results.
Answer
The simplified expression is \( 2\sqrt{2} + 4\sqrt{3} \).
Answer for screen readers
The simplified expression is ( 2\sqrt{2} + 4\sqrt{3} ).
Steps to Solve
- Simplify each square root separately
Start by simplifying the square roots:
For ( \sqrt{72} ):
- Factor ( 72 ) into ( 36 \times 2 ).
- Thus, ( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} ).
For ( \sqrt{32} ):
- Factor ( 32 ) into ( 16 \times 2 ).
- So, ( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} ).
For ( \sqrt{48} ):
- Factor ( 48 ) into ( 16 \times 3 ).
- Hence, ( \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} ).
- Combine the simplified results
Now that we have simplified each square root, we can substitute back into the expression:
$$ \sqrt{72} - \sqrt{32} + \sqrt{48} = 6\sqrt{2} - 4\sqrt{2} + 4\sqrt{3} $$
Now combine the like terms (the terms involving ( \sqrt{2} )):
- Combine ( 6\sqrt{2} - 4\sqrt{2} = 2\sqrt{2} ).
Thus, the expression simplifies to:
$$ 2\sqrt{2} + 4\sqrt{3} $$
The simplified expression is ( 2\sqrt{2} + 4\sqrt{3} ).
More Information
This expression combines multiple square roots into a simpler form, which is helpful in many mathematical contexts, such as solving equations or when dealing with geometry problems involving roots.
Tips
- Ignoring the negative sign: It's easy to forget the negative when combining terms. Always pay attention to the signs of the coefficients in front of the square roots.
- Not simplifying correctly: Ensure that you apply the square root rules properly when factoring the numbers.
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