What is the following product?
[tex]$(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$[/tex]
A. [tex]$2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$[/tex]
B. [tex]$ \sqrt{14}-6+\sqrt{7}$[/tex]
C. [tex]$ \sqrt{26}+\sqrt{21}-\sqrt{15}-\sqrt{10}$[/tex]
D. [tex]$2 \sqrt{42}-\sqrt{21}$[/tex]
Answer (2)
To calculate the product ( 14 − 3 ) ( 12 + 7 ) , we use the distributive property to expand and simplify it to 2 42 + 7 2 − 6 − 21 . This matches with option A from the multiple-choice answers provided. Therefore, the correct answer is option A.
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Expand the product ( 14 − 3 ) ( 12 + 7 ) using the distributive property.
Simplify each term in the expanded expression, such as 14 × 12 = 2 42 and 3 × 12 = 6 .
Combine the simplified terms to get 2 42 + 7 2 − 6 − 21 .
Compare the result with the given expressions to find the match: 2 42 + 7 2 − 6 − 21 .
Explanation
Expanding the Product We are asked to find the product of ( 14 − 3 ) ( 12 + 7 ) and determine which of the given expressions is equal to it. Let's first expand the product.
Applying the Distributive Property Expanding the product ( 14 − 3 ) ( 12 + 7 ) using the distributive property (also known as the FOIL method), we get:
14 × 12 + 14 × 7 − 3 × 12 − 3 × 7
Now, let's simplify each term:
Simplifying Each Term
14 × 12 = 14 × 12 = 168 = 4 × 42 = 2 42
14 × 7 = 14 × 7 = 2 × 7 × 7 = 7 2
3 × 12 = 3 × 12 = 36 = 6
3 × 7 = 3 × 7 = 21
Combining the Simplified Terms Substituting these simplified terms back into the expanded expression, we have:
2 42 + 7 2 − 6 − 21
Now, we compare this simplified expression with the four given expressions to find the matching one.
Finding the Matching Expression Comparing our simplified expression 2 42 + 7 2 − 6 − 21 with the given expressions:
2 42 + 7 2 − 6 − 21 - This matches our simplified expression.
14 − 6 + 7
26 + 21 − 15 − 10
2 42 − 21
Final Answer The simplified expression 2 42 + 7 2 − 6 − 21 matches the first given expression. Therefore, the product ( 14 − 3 ) ( 12 + 7 ) is equal to 2 42 + 7 2 − 6 − 21 .
Examples
Understanding how to simplify radical expressions and multiply them is useful in various fields, such as physics and engineering, when dealing with lengths, areas, or volumes that involve square roots. For example, when calculating the diagonal of a rectangle with sides 14 − 3 and 12 + 7 , you would need to multiply these expressions. Simplifying such expressions allows for more accurate and manageable calculations in real-world applications.
