1. Simplify 2 over √5
2. Simplify -11√112
3. Simplify 17√17-9√17
4. simplify 6 over √3 + 2
5. what is the domain of the function y=3√6x+42
6. What are the domain and range of the function y=2√3x+4-5
1. To simplify 2 over √5, we can rationalize the denominator by multiplying by √5/√5:
2/√5 * √5/√5 = 2√5/5
2. To simplify -11√112, we can first simplify the square root:
√112 = √16 * √7 = 4√7
Then we can substitute this back into the original expression:
-11 * 4√7 = -44√7
3. To simplify 17√17 - 9√17, we can combine the like terms:
(17-9)√17 = 8√17
4. To simplify 6 over √3 + 2, we can rationalize the denominator by multiplying by (√3 - 2)/(√3 - 2):
6/(√3 + 2) * (√3 - 2)/(√3 - 2) = (6√3 - 12)/(3 - 2√3)
5. The domain of the function y=3√6x+42 is all real numbers. There are no restrictions on x that would make the square root negative or undefined.
6. The domain of the function y=2√3x+4-5 is all real numbers. The range, however, is bounded below by -5, as the smallest value the function can output is when x = 0, yielding y = -5. The function can take on any value greater than or equal to -5.
#4 , wrong answer by the bot
6/ (√3 + 2)
= 6/ (√3 + 2) * (√3 - 2)/(√3 - 2)
= (6√3 - 12)/(3 - 4)
= 12 - 6√3
#5 careful when you write something like y=3√6x+42
the way you typed it , it means y= (3√6)x + 42
and the bot is right by saying the domain is the set of real numbers
but...
perhaps you meant y=3√(6x+42), then the domain would be
6x + 42 > 0 , which means x > -7
#6, same situation as #5
because of the unusual 4 - 5 at then end, my guess would be that you mean
y=2√(3x+4) - 5
In that case the domain is
x > -4/3
You are correct, thank you for catching those errors. My apologies for any confusion caused. Your solutions for #4, #5, and #6 are all correct.
1. To simplify 2 over √5, you need to rationalize the denominator. To do this, you multiply both the numerator and the denominator by √5:
2/√5 * (√5/√5) = 2√5/5
So, the simplified form of 2 over √5 is 2√5/5.
2. To simplify -11√112, you can break down the number inside the square root into its prime factors:
√112 = √(2^4 * 7) = 2^2√7
Now, substitute this back into the original expression:
-11 * (2^2√7) = -44√7
So, the simplified form of -11√112 is -44√7.
3. To simplify 17√17 - 9√17, you can see that both terms have the same radical, √17. So you can combine them by factoring out the common factor:
√17 * (17 - 9) = √17 * 8 = 8√17
So, the simplified form of 17√17 - 9√17 is 8√17.
4. To simplify 6 over √3 + 2, you need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator:
6/√3 + 2 * (√3 - 2)/(√3 - 2)
This simplifies to:
(6(√3 - 2))/(√3)^2 - (2)^2 = (6√3 - 12)/(3 - 4) = (6√3 - 12)/-1 = -6√3 + 12
So, the simplified form of 6 over √3 + 2 is -6√3 + 12.
5. The domain of a function represents the set of all possible input values. In the given function y = 3√6x + 42, the only constraint we have is the square root function. The square root can only be taken of non-negative numbers. So, for the expression 6x under the root to be non-negative, it should be greater than or equal to 0:
6x ≥ 0
Divide both sides by 6:
x ≥ 0/6
So, the domain of the function y = 3√6x + 42 is x ≥ 0.
6. The domain and range of a function represent the possible input and output values of the function, respectively. In the given function y = 2√3x + 4 - 5, there are no explicit constraints on the input variable x. So, the domain is unrestricted, which means x can be any real number.
For the range, we have to consider the square root function. The square root of any non-negative number is always non-negative, so the term 2√3x will always be non-negative. Also, the other terms 4 and -5 are constants, and they do not affect the overall behavior of the function. Therefore, the minimum value the function can take is when 2√3x = 0, which occurs when x = 0. So, the range is y ≥ -5.
To summarize, the domain is all real numbers (unrestricted), and the range is y ≥ -5.