Answer :
Certainly! To expand the expression \((1+\sqrt{2})(3-\sqrt{2})\) and give the answer in the form \(a + b\sqrt{2}\), we will proceed step-by-step using the distributive property.
First, let's distribute each term in the first binomial \((1+\sqrt{2})\) by each term in the second binomial \((3-\sqrt{2})\):
[tex]\[ (1 + \sqrt{2})(3 - \sqrt{2}) = 1 \cdot 3 + 1 \cdot (-\sqrt{2}) + \sqrt{2} \cdot 3 + \sqrt{2} \cdot (-\sqrt{2}) \][/tex]
Now, let's evaluate each term individually:
1. Multiply the constants:
[tex]\[ 1 \cdot 3 = 3 \][/tex]
2. Multiply the constant by the negative square root term:
[tex]\[ 1 \cdot -\sqrt{2} = -\sqrt{2} \][/tex]
3. Multiply the square root term by the constant:
[tex]\[ \sqrt{2} \cdot 3 = 3\sqrt{2} \][/tex]
4. Multiply the square root terms:
[tex]\[ \sqrt{2} \cdot (-\sqrt{2}) = -(\sqrt{2})^2 = -2 \][/tex]
Next, combine all these terms together:
[tex]\[ 3 + (-\sqrt{2}) + 3\sqrt{2} + (-2) \][/tex]
Now, let's group the integer terms and the square root terms:
1. Combine the integer terms:
[tex]\[ 3 + (-2) = 1 \][/tex]
2. Combine the square root terms:
[tex]\[ -\sqrt{2} + 3\sqrt{2} = 2\sqrt{2} \][/tex]
So, the expanded expression, combining both integer and square root parts, is:
[tex]\[ 1 + 2\sqrt{2} \][/tex]
Thus, the expression \((1+\sqrt{2})(3-\sqrt{2})\) in the form \(a + b\sqrt{2}\) is:
[tex]\[ \boxed{1 + 2\sqrt{2}} \][/tex]
First, let's distribute each term in the first binomial \((1+\sqrt{2})\) by each term in the second binomial \((3-\sqrt{2})\):
[tex]\[ (1 + \sqrt{2})(3 - \sqrt{2}) = 1 \cdot 3 + 1 \cdot (-\sqrt{2}) + \sqrt{2} \cdot 3 + \sqrt{2} \cdot (-\sqrt{2}) \][/tex]
Now, let's evaluate each term individually:
1. Multiply the constants:
[tex]\[ 1 \cdot 3 = 3 \][/tex]
2. Multiply the constant by the negative square root term:
[tex]\[ 1 \cdot -\sqrt{2} = -\sqrt{2} \][/tex]
3. Multiply the square root term by the constant:
[tex]\[ \sqrt{2} \cdot 3 = 3\sqrt{2} \][/tex]
4. Multiply the square root terms:
[tex]\[ \sqrt{2} \cdot (-\sqrt{2}) = -(\sqrt{2})^2 = -2 \][/tex]
Next, combine all these terms together:
[tex]\[ 3 + (-\sqrt{2}) + 3\sqrt{2} + (-2) \][/tex]
Now, let's group the integer terms and the square root terms:
1. Combine the integer terms:
[tex]\[ 3 + (-2) = 1 \][/tex]
2. Combine the square root terms:
[tex]\[ -\sqrt{2} + 3\sqrt{2} = 2\sqrt{2} \][/tex]
So, the expanded expression, combining both integer and square root parts, is:
[tex]\[ 1 + 2\sqrt{2} \][/tex]
Thus, the expression \((1+\sqrt{2})(3-\sqrt{2})\) in the form \(a + b\sqrt{2}\) is:
[tex]\[ \boxed{1 + 2\sqrt{2}} \][/tex]
