Question
(4 - 3square root 3)(4 + square root 21)
Answers
GPT-4o mini
To simplify the expression \((4 - 3\sqrt{3})(4 + \sqrt{21})\), we'll use the distributive property (also known as the FOIL method for binomials).
\[
(4 - 3\sqrt{3})(4 + \sqrt{21}) = 4 \cdot 4 + 4 \cdot \sqrt{21} - 3\sqrt{3} \cdot 4 - 3\sqrt{3} \cdot \sqrt{21}
\]
Calculating each term:
1. \(4 \cdot 4 = 16\)
2. \(4 \cdot \sqrt{21} = 4\sqrt{21}\)
3. \(-3\sqrt{3} \cdot 4 = -12\sqrt{3}\)
4. \(-3\sqrt{3} \cdot \sqrt{21} = -3\sqrt{63} = -3\sqrt{9 \cdot 7} = -9\sqrt{7}\)
Now, combine all the terms:
\[
16 + 4\sqrt{21} - 12\sqrt{3} - 9\sqrt{7}
\]
So the expression \((4 - 3\sqrt{3})(4 + \sqrt{21})\) simplifies to:
\[
16 + 4\sqrt{21} - 12\sqrt{3} - 9\sqrt{7}
\]
This is the simplified form of the expression.
\[
(4 - 3\sqrt{3})(4 + \sqrt{21}) = 4 \cdot 4 + 4 \cdot \sqrt{21} - 3\sqrt{3} \cdot 4 - 3\sqrt{3} \cdot \sqrt{21}
\]
Calculating each term:
1. \(4 \cdot 4 = 16\)
2. \(4 \cdot \sqrt{21} = 4\sqrt{21}\)
3. \(-3\sqrt{3} \cdot 4 = -12\sqrt{3}\)
4. \(-3\sqrt{3} \cdot \sqrt{21} = -3\sqrt{63} = -3\sqrt{9 \cdot 7} = -9\sqrt{7}\)
Now, combine all the terms:
\[
16 + 4\sqrt{21} - 12\sqrt{3} - 9\sqrt{7}
\]
So the expression \((4 - 3\sqrt{3})(4 + \sqrt{21})\) simplifies to:
\[
16 + 4\sqrt{21} - 12\sqrt{3} - 9\sqrt{7}
\]
This is the simplified form of the expression.