Multiply. Simplify if possible.
[tex]$(-1-\sqrt{10})(6+\sqrt{10})=$[/tex]
Answer (1)
We multiply the two binomials using the distributive property (FOIL method).
We combine the like terms.
The simplified result is − 16 − 7 10 .
Explanation
Understanding the problem We are asked to multiply two binomials: ( − 1 − 10 ) ( 6 + 10 ) . We will use the distributive property (also known as the FOIL method) to multiply the two binomials and then simplify the result by combining like terms.
Multiplying the binomials Let's multiply the binomials using the distributive property (FOIL method):
( − 1 − 10 ) ( 6 + 10 ) = − 1 ( 6 ) + − 1 ( 10 ) + − 10 ( 6 ) + − 10 ( 10 )
= − 6 − 10 − 6 10 − 10
Combining like terms Now, let's combine like terms:
= − 6 − 10 − 10 − 6 10
= − 16 − 7 10
Final result So, the simplified expression is − 16 − 7 10 .
Examples
Understanding how to multiply and simplify expressions like ( − 1 − 10 ) ( 6 + 10 ) is useful in various fields, such as physics and engineering, where you might encounter similar expressions when dealing with quadratic equations or complex numbers. For example, when calculating the eigenvalues of a matrix, you might need to simplify expressions involving square roots. This skill also helps in simplifying algebraic expressions in calculus and other advanced mathematics courses. Mastering these basic algebraic manipulations is crucial for solving more complex problems in these areas.
