1. √(9y - 196) + √196 = √49
My work:
I squared all 3 roots and got
9y - 196 + 196 = 49
The - 196 + 196 cancels out and we are left with 9y = 49.
divide 49 by 9. = 49/9. I'm pretty sure this is wrong. Can you show me where I messed up?
2. √x(√9 - 4/3) = √25
I'm not quite sure where to start with this.
3. (4 - 3√2)^2
My work: Distributive property gives me 4^2 - 3^2√2^2
The √2^2 simplifies to 2, 4^2 is 16, 3^2 is 9, So we have 16-9x2 Multiply first, so 16 - 18 = -2.
Is this correct?
4. √(135*b^2*c^3*d) * √(5b^2*d
Okay, so I broke this down like this: √135 * √b^2 * √c^2 * √c * √d * √5 * √b^2 * √d
Both the √b^2's and the √c^2 simplify down to b,b, and c
so now we have b^2c√(135 * 5cd^2) Simplifies to b^2cd√(5^2 * 3^2 * 3) which simplifies to 15b^2cd√3
Is this correct? If not can you tell me where I went wrong?
Thank you!
If a+b = c you cannot say that a^2+b^2 = c^2
√(9y - 196) + √196 = √49
√(9y - 196) + 14 = 7
√(9y - 196) = -7
But, √N is positive. So, you cannot have √(9y - 196) = -7
√x(√9 - 4/3) = √25
√x(3 - 4/3) = 5
√x(5/3) = 5
√x = 3
x = 9
(4 - 3√2)^2
recall that (a-b)^2 = a^2-2ab+b^2, so we have
16 - 24√2 + 18
44 - 24√2
almost right - you lost a √c
√(135*b^2*c^3*d) * √(5b^2*d)
√(135*5) √(b^2*b^2) √c^3 √(d*d)
15√3 b^2 c√c d
15b^2cd √(3c)
I have a couple questions about # 1.
Why can't it be a^2 + B^2 = c^2?
I think that's what I'm taught in class. Also, if you square everything, isn't still going to come out the same? If you do something to the entire expression, it doesn't change the value, right? If your conclusion is correct, what would I put for that answer? "No solution"? "No real solution"?
On #3. Would I do 44-24 = 20√2.. or?
For #1: When I get √(9y - 196) = -7 can I square each side to get 9y - 196 = 49? Then I can add 196 to both sides. It cancels out on the left side and I have 9y = 245. Then I can divide both sides by 9. I would get y = 27.2. Would that be correct?
You say that I can't use a^2 + b^2 = c^2, but that formula is called the Pythagorean theorem. I was instructed to use that formula. Am I misunderstanding what you're saying?
Let's go through each question and identify any mistakes:
1. √(9y - 196) + √196 = √49
You correctly squared all three roots and got 9y - 196 + 196 = 49. However, you made a mistake when you canceled out -196 and +196. It's important to remember that the square root function is not distributive over addition or subtraction. So the equation should be 9y - 196 + 196 = 49.
When you simplify this equation, you get 9y = 49. To solve for y, you divide both sides of the equation by 9, which gives y = 49/9. So your answer is correct.
2. √x(√9 - 4/3) = √25
To solve this equation, you can start by simplifying each side individually. √9 is equal to 3, and √25 is equal to 5. So the equation becomes √x(3 - 4/3) = 5.
Next, simplify the expression within the parentheses: 3 - 4/3. To do that, you can find a common denominator, which is 3. Rewrite 3 as 9/3, and the equation becomes √x(9/3 - 4/3) = 5.
Now, subtract the fractions within the parentheses: 9/3 - 4/3 = 5/3. So the equation becomes √x * (5/3) = 5.
To isolate √x, divide both sides of the equation by 5/3: √x = 5 / (5/3). When you divide by a fraction, you can multiply by its reciprocal. So the equation becomes √x = 5 * (3/5).
Multiply the numbers: √x = 15/5. Simplify the fraction: √x = 3.
To solve for x, square both sides of the equation: (√x)^2 = 3^2. This gives x = 9. So your final answer is x = 9.
3. (4 - 3√2)^2
You correctly applied the distributive property and squared each term. The first term (4) squared is 16. The second term (-3√2) squared is (-3√2) * (-3√2). Remember that (a - b)^2 is equal to a^2 - 2ab + b^2.
So, the equation becomes 16 - 2 * (4 * 3√2) + (3√2)^2. Simplify further to 16 - 24√2 + 9 * 2.
Now, multiply: 16 - 24√2 + 18.
Combine like terms: 34 - 24√2.
Therefore, the correct answer is 34 - 24√2.
4. √(135b^2c^3d) * √(5b^2d)
You correctly broke down the expression into separate square roots. However, there was a mistake while simplifying.
Starting with √(135b^2c^3d) * √(5b^2d), simplify each term:
√(135b^2c^3d) becomes √(9 * 15 * (b * b) * (c * c * c) * d) and √(5b^2d) remains the same.
Now, extract perfect square factors from under the square roots. For example, 9 is a perfect square, so we can take its square root and move it outside the square root sign: 9 becomes 3.
Applying this to both square roots, we get 3 * √(15 * (b * b) * (c * c * c) * d) * √(5b^2d).
The √(b^2) simplifies to b, so we have:
3 * b * c * √(15 * (b * c * c * c * d)) * √(5 * d).
Combining like terms, we have:
3b^2c√(15 * b * c * c * c * d * 5 * d).
Simplifying further, we have:
3b^2c√(75b^2c^3d^2).
Finally, you can simplify the square root of 75 by breaking it down into perfect square factors. The square root of 75 is √(25 * 3), which further simplifies to 5√3.
So the final answer is:
15b^2c^2d√(3d).
Overall, your approach was correct for all the questions. Just be cautious with canceling out terms under the square root and remember that the square root function is not distributive over addition or subtraction.