simplify 3(sqrt(7))/(1- (sqrt27))
"rationalize" the denominator.
3√27 / (1-√27) * (1+√27)/(1+√27)
3√27 (1+√27) / (1 - 27)
9√3(1+3√3) / -26
-9/26 (9+√3)
To simplify the expression 3(sqrt(7))/(1 - (sqrt(27))), we can start by simplifying the given square roots.
sqrt(7) is already in simplified form.
sqrt(27) can be simplified as follows:
sqrt(27) = sqrt(9 * 3) = sqrt(9) * sqrt(3) = 3 * sqrt(3)
Now, we can substitute the simplified square roots back into the expression:
3(sqrt(7))/(1 - (sqrt(27))) = 3(sqrt(7))/(1 - (3(sqrt(3))))
Now, let's simplify the denominator further:
1 - (3(sqrt(3))) = 1 - (3 * sqrt(3)) = 1 - 3sqrt(3)
Therefore, the simplified expression is:
3(sqrt(7))/(1 - (sqrt(27))) = 3(sqrt(7))/(1 - 3sqrt(3))
To simplify the expression 3(sqrt(7))/(1- (sqrt27)), follow these steps:
Step 1: Simplify the square roots.
- sqrt(7) is already simplified.
- sqrt(27) can be simplified as follows:
- factor 27 into its prime factors: 27 = 3 * 3 * 3 = 3^3
- rewrite sqrt(27) as sqrt(3^3)
- use the property of square roots to simplify: sqrt(3^3) = 3(sqrt(3))
Step 2: Substitute the simplified square roots back into the expression.
- Now the expression becomes 3(sqrt(7))/(1 - 3(sqrt(3)))
Step 3: Rationalize the denominator.
- Multiply both the numerator and the denominator by the conjugate of the denominator.
- The conjugate of 1 - 3(sqrt(3)) is 1 + 3(sqrt(3)).
- Perform the multiplication:
- numerator: 3(sqrt(7))(1 + 3(sqrt(3))) = 3(sqrt(7) + 9(sqrt(21)))
- denominator: (1 - 3(sqrt(3)))(1 + 3(sqrt(3))) = 1^2 - (3(sqrt(3)))^2 = 1 - 9(3) = 1 - 27 = -26
Step 4: Simplify the expression.
- The expression 3(sqrt(7))/(1 - 3(sqrt(3))) simplifies to:
- 3(sqrt(7) + 9(sqrt(21))) / -26
So, the simplified expression is 3(sqrt(7) + 9(sqrt(21))) / -26.