Multiply 93 −−√ ⋅67–√ . Give an exact answer.(1 point)
To multiply $\sqrt{93}$ and $\sqrt{67}$, we can write them as a single square root:
$\sqrt{93} \cdot \sqrt{67} = \sqrt{93 \cdot 67} = \sqrt{6231}$
Therefore, the exact answer is $\sqrt{6231}$.
To multiply 93 −−√ ⋅67–√, we can simplify each of the square roots first:
√93 = √(9 * 10 + 3) = √(9) * √(10) + √(3) = 3√10 + √3
√67 = √(6 * 10 + 7) = √(6) * √(10) + √(7) = 2√10 + √7
Now, we can multiply these two expressions:
(3√10 + √3) * (2√10 + √7)
Using the distributive property, we can expand this expression:
3√10 * 2√10 + 3√10 * √7 + √3 * 2√10 + √3 * √7
Simplifying each term:
6 * 10 + 3√10√7 + 2√10√3 + √3√7
Combine like terms:
60 + 3√70 + 2√30 + √21
Therefore, the exact answer of the multiplication is 60 + 3√70 + 2√30 + √21.
To multiply these two square root expressions, you can use the properties of square roots. Recall that the square root of a product is equal to the product of the square roots. In this case, we have:
\[ \sqrt{93} \cdot \sqrt{67} \]
To simplify further, we need to find the prime factorization of each number under the square root.
The prime factorization of 93 is:
\[ 93 = 3 \cdot 31 \]
The prime factorization of 67 is:
\[ 67 = 67 \]
Applying the property of square roots, we can rewrite the expression as:
\[ \sqrt{93} \cdot \sqrt{67} = \sqrt{3 \cdot 31} \cdot \sqrt{67} \]
Using the property of square roots again, we can split the product into two separate square roots:
\[ \sqrt{3 \cdot 31} \cdot \sqrt{67} = \sqrt{3} \cdot \sqrt{31} \cdot \sqrt{67} \]
Now, we can simplify each square root separately:
\[ \sqrt{3} \cdot \sqrt{31} \cdot \sqrt{67} = \sqrt{3} \cdot \sqrt{31} \cdot \sqrt{67} \]
So, the exact answer to the multiplication is \( \sqrt{3} \cdot \sqrt{31} \cdot \sqrt{67} \).