I am not sure how to do these.
1. Simplify 4√6/√30 by rationalizing the denominator. Show your work.
2. Simplify (2√5 + 3√7)^2 Show your work. Justify each step.
4 sqrt 6
----------
sqrt (6*5)
= 4/sqrt 5
* sqrt 5/sqrt 5
= (4/5) sqrt 5
Sure, I can help you with those!
1. To simplify 4√6/√30 by rationalizing the denominator, we need to get rid of the square root in the denominator.
Step 1: Let's start by simplifying the square roots in both the numerator and denominator.
- The square root of 6 can be simplified as √6 = √(2*3) = √2√3.
- The square root of 30 can be simplified as √30 = √(2*3*5) = √2√3√5.
Step 2: Now, substitute these simplified square roots back into the original expression.
- The expression becomes 4√2√3 / √2√3√5.
Step 3: Next, we can cancel out the common square roots in the numerator and denominator.
- This gives us 4/√5.
Step 4: To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of √5, which is also √5.
- Multiplying 4/√5 by √5/√5 gives us (4√5)/(√5 * √5) = (4√5)/5.
So, the simplified expression 4√6/√30 after rationalizing the denominator is (4√5)/5.
2. To simplify (2√5 + 3√7)^2, we need to expand the square.
Step 1: Let's first write the expression using the distributive property:
- (2√5 + 3√7)(2√5 + 3√7).
Step 2: Now, we can expand using the FOIL method (First, Outer, Inner, Last):
- First: (2√5 * 2√5) = 4 * 5 = 20.
- Outer: (2√5 * 3√7) = 6√(5*7) = 6√35.
- Inner: (3√7 * 2√5) = 6√(7*5) = 6√35.
- Last: (3√7 * 3√7) = 9 * 7 = 63.
Step 3: Combine the terms obtained from the FOIL method:
- 20 + 6√35 + 6√35 + 63.
Step 4: Simplify the expression:
- Combining like terms, we have 83 + 12√35.
So, the simplified expression for (2√5 + 3√7)^2 is 83 + 12√35.
Sure! I can help you with both of these problems.
1. To simplify 4√6/√30 by rationalizing the denominator, we need to get rid of the square root in the denominator.
To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of √30, which is √30.
Then we have:
(4√6/√30) * (√30/√30)
Multiplying the numerators and the denominators, we get:
(4√6 * √30) / (√30 * √30)
Simplifying further, we have:
(4√(6 * 30)) / 30
Now, we can simplify the expression inside the square root:
(4√(180)) / 30
Since 180 is a perfect square, we can simplify further:
(4 * √(36 * 5)) / 30
(4 * 6√5) / 30
Finally, we can simplify this fraction by canceling out the common factors:
(24√5) / 30
Reducing the fraction to its simplest form, we get:
(12√5) / 15
Therefore, the simplified expression is 12√5/15.
2. To simplify (2√5 + 3√7)^2, we need to square both terms inside the parentheses and then simplify the resulting expression.
Using the FOIL method, we can expand the squared expression as follows:
(2√5 + 3√7) * (2√5 + 3√7)
First, we multiply the first terms:
(2√5 * 2√5)
= 4 * 5
= 20
Next, we multiply the two cross terms:
(2√5 * 3√7) + (3√7 * 2√5)
= 6√35 + 6√35
= 12√35
Finally, we multiply the last terms:
(3√7 * 3√7)
= 9 * 7
= 63
Now, we add up the three results:
20 + 12√35 + 63
= 83 + 12√35
Therefore, the simplified expression is 83 + 12√35.
This is how you can simplify both of these expressions. Let me know if you have any other questions!