Sure! Let's simplify the given radical expression step by step. The expression is:
[tex]\[ (9 + 4 \sqrt{7})(2 \sqrt{7} - 5) \][/tex]
We will use the distributive property of multiplication to expand this product. This means we will multiply each term in the first expression by each term in the second expression. Let's do this:
1. Multiply [tex]\(9\)[/tex] by [tex]\(2 \sqrt{7}\)[/tex]:
[tex]\[ 9 \cdot 2 \sqrt{7} = 18 \sqrt{7} \][/tex]
2. Multiply [tex]\(9\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[ 9 \cdot (-5) = -45 \][/tex]
3. Multiply [tex]\(4 \sqrt{7}\)[/tex] by [tex]\(2 \sqrt{7}\)[/tex]:
[tex]\[ 4 \sqrt{7} \cdot 2 \sqrt{7} = 8 \cdot 7 = 56 \][/tex]
(Note that [tex]\(\sqrt{7} \cdot \sqrt{7} = 7\)[/tex])
4. Multiply [tex]\(4 \sqrt{7}\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[ 4 \sqrt{7} \cdot (-5) = -20 \sqrt{7} \][/tex]
Now we combine these products together:
[tex]\[ 18 \sqrt{7} - 45 + 56 - 20 \sqrt{7} \][/tex]
Next, we separate the constant terms from the terms with [tex]\(\sqrt{7}\)[/tex]:
[tex]\[ (18 \sqrt{7} - 20 \sqrt{7}) + (-45 + 56) \][/tex]
Combine the like terms:
1. For the [tex]\(\sqrt{7}\)[/tex] terms:
[tex]\[ 18 \sqrt{7} - 20 \sqrt{7} = -2 \sqrt{7} \][/tex]
2. For the constant terms:
[tex]\[ -45 + 56 = 11 \][/tex]
Putting it all together, we get:
[tex]\[ 11 - 2 \sqrt{7} \][/tex]
Therefore, the simplified form of the given radical expression is:
[tex]\[ 11 - 2 \sqrt{7} \][/tex]
