How do I work this to get the answer because I am stuck. Help, Help, please anyone. Ihope that you are able to understand what I am asking.
3 SQRT10 over SQRT15
2 over 5+SQRT3
You have to get the square roots out of the denominator by multipling the top and bottom by (for the first on) the SQRT15. And the second one by (5-SQRT3). Then you should be able to simplify. I hope that helps a little!
1. 3 sqrt10 / sqrt15 =
Multiply numerator and denominator by sqrt15:
3sqrt10*sqrt15 /15 = 3sqrt(10*15)/15 =
3sqrt(25*6) / 15 = 3*5sqrt(6) / 15 =
sqrt6. = 2.45
2. 2 / (5 + sqrt3) =
Multiply num. and den. by (5 - sqrt3)
2(5 - sqrt3) / (5 + sqrt3) (5 - sqrt3)
= (10 - 2sqrt3) / 22 = (5 - sqrt3) / 11
= 0.297
Of course, I'm here to help! Let's break down each question and explain how to find the answers step by step.
Question 1: 3√10/√15
To simplify this expression, we need to rationalize the denominator, which means eliminating the square roots in the denominator.
Step 1: Simplify the square roots in the denominator
To simplify √15, we can break it down into its prime factors. The factors of 15 are 3 and 5. Since we want to simplify the square root, we can write it as √(3 × 5).
Step 2: Rationalize the denominator
To rationalize the denominator, we want to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of √(3 × 5) is √(3 × 5) because multiplying a square root by itself eliminates the square root.
So, our expression becomes:
(3√10/√15) × (√(3 × 5)/√(3 × 5))
Step 3: Simplify the expression
Multiplying the numerators and denominators together, we get:
(3√10 × √(3 × 5))/(√(3 × 5) × √(3 × 5))
Combining the square roots in the numerator, we have:
(3√(10 × 3 × 5))/(√(3 × 5) × √(3 × 5))
Multiplying under the square root sign, we get:
(3√150)/(√15 × √15)
Simplifying the square root of 150, we have:
(3 × √(5 × 5 × 2))/(√15 × √15)
Finally, we can simplify the expression to:
(3 × 5√2)/(√15 × √15)
Step 4: Further Simplification
Since √15 × √15 = 15, we have:
(3 × 5√2)/15
And simplifying the expression further:
(15√2)/15
Canceling out the common factor of 15, we get the final answer:
√2
Therefore, the simplified expression is √2.
Question 2: 2/(5 + √3)
To simplify this expression, we also need to rationalize the denominator.
Step 1: Rationalize the denominator
To get rid of the square root in the denominator, we want to multiply both the numerator and the denominator by the conjugate of the denominator, which is (5 - √3).
So, our expression becomes:
(2/(5 + √3)) × ((5 - √3)/(5 - √3))
Step 2: Simplify the expression
Multiplying the numerators and denominators together, we get:
(2 × (5 - √3))/((5 + √3) × (5 - √3))
Expanding the denominator, we have:
(2 × (5 - √3))/(5^2 - (√3)^2)
Which simplifies to:
(2 × (5 - √3))/(25 - 3)
Further simplifying the expression, we get:
(2 × (5 - √3))/22
And rearranging the terms:
(10 - 2√3)/22
Now, we can simplify the expression further:
(5 - √3)/11
Therefore, the simplified expression is (5 - √3)/11.
I hope this explanation helps you understand how to work through these types of problems! Let me know if you have any more questions.