1. Enter your answer and show all the steps that you use to solve this problem in the space provided.
Simplify
4√6/√30
by rationalizing the denominator. Show your work.
2. Enter your answer and show all the steps that you use to solve this problem in the space provided.
Simplify
(2√5+3√7)^2
. Show your work. Justify each step.
1,
Multiply numerator and denominator by √30
4 ∙ √6 ∙ √30 / √30 ∙ √30 = 4 ∙ √6 ∙ √30 / 30 = 4 ∙ √ ( 6 ∙ 30 ) / 30 =
4 ∙ √180 / 30 = 4 ∙ √ ( 36 ∙ 5 ) / 30 = 4 ∙ √ ( 6² ∙ 5 ) / 30 =
4 ∙ 6 ∙ √ 5 / 30 = 6 ∙ 4 ∙ √ 5 / 6 ∙ 5 = 4 ∙ √ 5 / 5 =
4 ∙ √ 5 / √ 5 ∙ √ 5 = 4 / √ 5
2.
Use square of sum formula:
( a + b )² = a² + 2 ∙ a ∙ b + b²
( 2 √5 + 3 √7 )² = ( 2 √5 )² + 2 ∙ 2 √5 ∙ 3 √7 + ( 3 √7 )² =
2² ∙ (√5 )² + 12 √5 ∙ √7 + 3² ∙ ( √7 )² = 4 ∙ 5 + 12 √ ( 5 ∙ 7 ) + 9 ∙ 7 =
20 + 12 √35 + 63 = 83 + 12 √35
OH MY GOSH THANK YOU SO MUCH!
1. To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator.
Let's start with the expression: 4√6/√30
The conjugate of √30 is -√30, so we will multiply both the numerator and denominator by -√30:
(4√6/√30) * (-√30/-√30)
Simplifying the expression, we get:
-4√6 * √30 / (-√30 * -√30)
Next, we multiply the terms within the numerator:
-4 * 6 * √30 * √30 / (-√30 * -√30)
Simplifying further:
-24 * 30 / (-√30 * -√30)
Multiplying the numbers:
-720 / (-√30 * -√30)
Finally, simplifying the square root:
-720 / (√30 * √30)
The square root of 30 is 5√2, so the expression becomes:
-720 / (5√2 * 5√2)
Simplifying:
-720 / (25 * 2)
-720 / 50
The final simplified expression is:
-14.4
2. We have the expression: (2√5 + 3√7)^2
To solve this, we'll use the formula (a + b)^2 = a^2 + 2ab + b^2.
Let's substitute a with 2√5 and b with 3√7:
(2√5 + 3√7)^2 = (2√5)^2 + 2(2√5)(3√7) + (3√7)^2
Simplifying the square terms:
(4*5) + 2(2√5)(3√7) + (9*7)
20 + 2(2√5)(3√7) + 63
Multiplying the numbers:
20 + 2(6√35) + 63
Next, we distribute the 2:
20 + 12√35 + 63
Adding the numbers:
83 + 12√35
Therefore, the simplifed expression is:
83 + 12√35
Question 1:
To simplify the expression 4√6/√30, we need to rationalize the denominator.
Step 1: Simplify the numerator and denominator separately.
The square root of 6 can be simplified as √(6) = √(2 * 3) = √2 * √3.
Similarly, the square root of 30 can be simplified as √(30) = √(2 * 3 * 5) = √2 * √3 * √5.
Step 2: Rewrite the expression with the simplified radicals in the numerator and denominator.
4√6/√30 becomes 4(√2 * √3)/(√2 * √3 * √5).
Step 3: Cancel out the common factors between the numerator and denominator.
We can cancel out the √2 and √3 terms in the numerator and denominator, leaving us with:
(4 * 1)/(1 * √5) = 4/√5.
Therefore, the simplified form of 4√6/√30 is 4/√5.
Question 2:
To simplify the expression (2√5 + 3√7)^2, we can use the distributive property and the rules for multiplying binomials.
Step 1: Expand the expression using the FOIL method.
(2√5 + 3√7)^2 = (2√5 + 3√7)(2√5 + 3√7)
= (2√5)(2√5) + (2√5)(3√7) + (3√7)(2√5) + (3√7)(3√7)
= 4√5 * √5 + 6√5 * √7 + 6√7 * √5 + 9√7 * √7
= 4√(5 * 5) + 6√(5 * 7) + 6√(7 * 5) + 9√(7 * 7)
= 4√25 + 6√35 + 6√35 + 9√49
= 4√25 + 12√35 + 9√49
= 4 * 5 + 12√35 + 9 * 7
= 20 + 12√35 + 63
= 83 + 12√35.
Therefore, the simplified form of (2√5 + 3√7)^2 is 83 + 12√35.