Multiply 93 −−√ ⋅67–√ . Give an exact answer
To multiply the expressions, we need to multiply the square roots as well as the numbers outside the square roots.
Starting with the numbers outside the square roots:
93 * 67 = 6219
Now, let's multiply the square roots:
√93 * √67 = √(93 * 67)
To multiply 93 and 67 inside the square roots, we first prime factorize each number:
93 = 3 * 31
67 = 67
Multiplying the prime factors:
93 * 67 = (3 * 31) * 67 = 3 * 31 * 67
Now, let's simplify the square root:
√(93 * 67) = √(3 * 31 * 67)
Since there are no square factors left, we can't simplify further.
Putting it all together:
93 −−√ ⋅67–√ = √(93 * 67) = √(3 * 31 * 67) = (√3)(√31)(√67) = (√3)(√31)(√67)
So the exact answer is (√3)(√31)(√67).
Divide 265√5√ . Provide an exact answer.
To divide the expression, we need to divide both the numbers outside the square root and the square root terms.
Starting with the numbers outside the square roots:
265 ÷ 5 = 53
Now, let's divide the square roots:
√(265) ÷ √(5)
To divide the square roots, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator (which is √5):
(√(265) ÷ √(5)) * (√(5) ÷ √(5)) = (√(265) * √(5)) ÷ (√(5) * √(5))
Simplifying the denominator:
(√(265) * √(5)) ÷ (√(5) * √(5)) = (√(265) * √(5)) ÷ 5
Now, let's simplify the square roots:
√(265) * √(5) = √(265 * 5) = √(1325)
So, the final expression is:
(√(265) * √(5)) ÷ 5 = √(1325) ÷ 5
The exact answer is √(1325) ÷ 5.
To solve this multiplication problem with square roots, we can perform the following steps:
Step 1: Simplify the square roots.
93 −−√ simplifies to √93.
67–√ simplifies to √67.
Step 2: Multiply the simplified square roots.
√93 * √67 = √(93 * 67)
Step 3: Multiply the numbers under the square root symbol.
√(93 * 67) = √(6231)
Step 4: Simplify the square root of the result.
The square root of 6231 cannot be simplified further (assuming we are not using any decimals or any approximations), so the final answer is √(6231).
To multiply two numbers, we need to multiply the numbers outside the square root sign and then multiply the numbers inside the square root sign separately.
Let's start with the numbers outside the square root sign: 93 and 67.
93 multiplied by 67 is 6,231.
Now, let's move on to the numbers inside the square root sign: √93 and √67.
To multiply these, we need to multiply the numbers inside the square root: 93 and 67.
√93 multiplied by √67 is equal to √(93 * 67).
Multiplying 93 and 67 gives us 6,231. So, √(93 * 67) simplifies to √6,231.
The exact answer is 6,231√6,231.