√27 x √243 x √12 / √125 x √18
that would be, assuming the usual carelessness with parentheses,
√(27*243*12 / 125*18) = 27/25 √30
or,
√(3^3*3^5*2^2*3 / 5^3*2*3^2)
= √(2^2*3^9 / 2*3^2*5^3)
= √(2*3^7/5^3)
= 3^3/5^2 √(2*3*5)
= 27/25 √30 or 27/5 √(6/5)
radical 27 = radical 9 times 3 so 3 radical 3
If you do this with all of the radicals, it makes the math a lot easier.
243 is 3 times 81 so 9 radical 3
12 = 4 times 3 so you have 2 radical 3
To simplify the expression √27 x √243 x √12 / √125 x √18, we can begin by simplifying each square root separately.
Let's simplify each term step by step:
√27 = √(9 x 3) = √9 x √3 = 3 x √3
√243 = √(81 x 3) = √81 x √3 = 9 x √3
√12 = √(4 x 3) = √4 x √3 = 2 x √3
√125 = √(25 x 5) = √25 x √5 = 5 x √5
√18 = √(9 x 2) = √9 x √2 = 3 x √2
Now we combine the simplified terms:
(3 x √3) x (9 x √3) x (2 x √3) / (5 x √5) x (3 x √2)
Multiplying the factors and rearranging:
(3 x 9 x 2) x (√3 x √3 x √3 x √2) / (5 x 3 x √5 x √2)
Simplifying the numbers and square roots:
54 x (√27 x √2) / (15 x √10)
Further simplification:
54 x (√(27 x 2)) / (15 x √10)
54 x (√54) / (15 x √10)
Now, we can simplify the square root of 54:
√54 = √(9 x 6) = √9 x √6 = 3 x √6
Substituting back into the expression:
54 x (3 x √6) / (15 x √10)
Now we can simplify further:
(54 x 3 x √6) / (15 x √10) = (162 x √6) / (15 x √10)
We can simplify the expression further by canceling out common factors:
(162 / 15) x (√6 / √10) = 10.8 x √(6 / 10) = 10.8 x √0.6
The final simplified expression is 10.8 x √0.6.
To simplify the expression √27 x √243 x √12 / √125 x √18, you can follow these steps:
Step 1: Simplify each square root individually.
√27 = √(3 x 3 x 3) = 3√3
√243 = √(3 x 3 x 3 x 3 x 3) = 9√3
√12 = √(3 x 2 x 2) = 2√3
√125 = √(5 x 5 x 5) = 5√5
√18 = √(2 x 3 x 3) = 3√2
Step 2: Substitute the simplified expressions back into the main expression.
(3√3)(9√3)(2√3) / (5√5)(3√2)
Step 3: Combine like terms in the numerator and denominator.
3 x 9 x 2 x √3 x √3 x √3 / 5 x √5 x √2
54√3 x √3 x √3 / 5√5 x √2
Step 4: Multiply the square roots in the numerator and denominator.
54 x 3 x 3 x 3 / 5 x √5 x √2
486 / 5√10
Step 5: Multiply the denominator by its conjugate to rationalize the denominator.
To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of 5√10 which is 5√10.
(486 / 5√10) x (5√10 / 5√10) = 2430√10 / 50
Step 6: Simplify the expression.
2430 / 50 = 48.6
Therefore, √27 x √243 x √12 / √125 x √18 simplifies to 48.6.