Multiply 93 −−√ ⋅67–√ . Give an exact answer.
To multiply these expressions with square roots, we need to simplify the square roots first.
√93 can be simplified.
Let's prime factorize 93:
93 = 3 * 31
Now, we pair the prime factors:
√93 = √(3 * 31)
Taking the square root of each pair:
√(3 * 31) = √3 * √31 = √3√31
Similarly, we can simplify √67:
67 is a prime number, so √67 is already in its simplest form.
Now we can multiply:
√93 * √67 = √3√31 * √67
√3 and √31 are both square roots and cannot be further simplified, so we leave them as they are.
√3√31 * √67 = √3 * √31 * √67 = √3 * √31 * √67
Thus, the exact answer is √3 * √31 * √67.
To multiply the expression (93-√2) by (67-√3), we can use the FOIL method.
FOIL stands for:
F - First: Multiply the first terms of each binomial.
O - Outer: Multiply the outer terms of each binomial.
I - Inner: Multiply the inner terms of each binomial.
L - Last: Multiply the last terms of each binomial.
Let's go step by step:
Step 1: Multiply the first terms:
93 * 67 = 6,231.
Step 2: Multiply the outer terms:
93 * -√3 = -93√3.
Step 3: Multiply the inner terms:
-√2 * 67 = -67√2.
Step 4: Multiply the last terms:
-√2 * -√3 = √6.
Now, let's combine the terms:
Step 5: Add up the products from steps 1 to 4:
6,231 + (-93√3) + (-67√2) + √6.
This will be the exact answer to the expression (93-√2) * (67-√3).
To multiply two numbers with square roots, you can use the property √(a) * √(b) = √(a * b). Let's apply this property to multiply 93 −−√ and 67–√.
First, we can simplify each square root individually. To simplify √93, we need to find the largest perfect square that divides 93. 93 is between 81 (9^2) and 100 (10^2). Thus, we can write 93 as √(9 * 10 + 3) = √(9) * √(10) + √(3). The square root of 9 is 3, so we have 3√(10) + √(3).
Similarly, for √67, 67 is between 64 (8^2) and 81 (9^2). We can write 67 as √(8 * 8 + 3) = √(8) * √(8) + √(3). The square root of 8 is 2√(2), so we have 2√(2) + √(3).
Now, let's multiply these two expressions: (3√(10) + √(3)) * (2√(2) + √(3)).
Using the distributive property, multiply each term in the first expression by each term in the second expression:
(3√(10)) * (2√(2)) + (3√(10)) * (√(3)) + (√(3)) * (2√(2)) + (√(3)) * (√(3)).
Simplifying each term, we get:
6 * √(20) + 3 * √(10) + 2 * √(6) + 3.
Therefore, the exact answer to the multiplication problem 93 −−√ ⋅ 67–√ is 6√(20) + 3√(10) + 2√(6) + 3.