Mathematics, 24.11.2020 03:00 kenken2583

\sqrt{53+20\sqrt{7}} can be written in the form a+b\sqrt{c}, where a, b, and c are integers and c has no factors which is a perfect square of any positive integer other than 1. Find a+b+c.

Answers: 1

Show answers

Answers

Answer from: charleswoodward438

=====================================================

Explanation:

Since the range of $y = \sqrt{x}$ is $y \ge 0$ , this means that $\sqrt{53+20\sqrt{7}}$ cannot be negative.

-----------------------------------

Let's set the given nested radical expression equal to the form a+b*sqrt(c), and then square both sides to see what happens

$\sqrt{53+20\sqrt{7}} = a+b\sqrt{c}\\\\\left(\sqrt{53+20\sqrt{7}}\right)^2 = \left(a+b\sqrt{c}\right)^2\\\\53+20\sqrt{7} = a^2+2ab\sqrt{c}+b^2c\\\\53+20\sqrt{7} = (a^2+b^2c)+2ab\sqrt{c}\\\\$

Equating the corresponding parts shows that

$53=a^2+b^2c \text{ and } 20\sqrt{7} = 2ab\sqrt{c}\\\\$

Therefore c = 7 must be the case to allow the radicands to match up.

-----------------------------------

Since c = 7, we can update the first equation like so

$53=a^2+b^2c\\\\53=a^2+b^2*7\\\\53=a^2+7b^2\\\\$

We'll come back to this later.

-----------------------------------

Focus on $20 = 2ab\sqrt{7}$

Note how 20 and 2ab are coefficients outside the root. So 20 = 2ab leads to ab = 10.

Here are all the ways to multiply to 10 where a,b are integers

1*10 = 10-1*(-10) = 102*5 = 10-2*(-5) = 10

So we could have these possibilities.

a = 1, b = 10a = 10, b = 1a = -10, b = -1a = -1, b = -10a = 2, b = 5a = 5, b = 2a = -5, b = -2a = -2, b = -5

If you try all of those a,b combinations with 53 = a^2 + 7b^2 , then you should find that a = 5 and b = 2 works. I'm skipping a bunch of steps here. All you basically do is plug and chug.

Due to symmetry, we could have a = -5 and b = -2 as another solution to 53 = a^2 + 7b^2 . However, if a < 0 and b < 0, then $a+b\sqrt{c}$ would be negative which contradicts the fact that $\sqrt{53+20\sqrt{7}}$ is positive. So we'll ignore any case where a,b are negative.

Therefore, the only combination that works is a = 5 and b = 2

-----------------------------------

Overall, we found that

a = 5b = 2c = 7

And therefore,

$\sqrt{53+20\sqrt{7}} = a+b\sqrt{c}\\\\\sqrt{53+20\sqrt{7}} = 5+2\sqrt{7}\\\\$

You can confirm this with your calculator by taking advantage of the idea "If x = y, then x-y = 0". In other words, if you subtract the left and right sides, you should get 0 as a result (the order of subtraction doesn't matter in this case). Don't forget to use parenthesis correctly, or you may not get 0. It might help to think of it as the form (x) - (y) = 0.

After we confirmed we have the correct a,b,c values, we simply add them up to get a+b+c = 5+2+7 = 14.

Answer from: Quest

the answer is b

step-by-step explanation:

Answer from: Quest

total amount of money that he should pay if he has not used his duty free exemption is $98

step-by-step explanation:

given a united states citizen returning to the states declares the following items at the customs office: 3 shirts at $8.50 each 2 dresses at $27.50 each 1 pair of gold cuff links at $17.50 per pair. we have to find how much he pay

cost of 3 shirts if one cost at $8.50 is $3\times 8.50=$25.5$

cost of 2 dresses if one cost at $27.50 is $2\times 27.50=$55$

cost of 1 pair of gold cuff at $17.50

so, total amount of money that he should pay if he has not used his duty free exemption is $25.5+$55+$17.50=$98

Another question on Mathematics

Mathematics, 21.06.2019 20:30

Wrote and expression for the difference of p and 4

Answers: 1

Answer

Mathematics, 21.06.2019 23:00

Complete the conditional statement. if a + 2 < b + 3, then a < b b < a a – b < 1 a < b + 1

Answers: 3

Answer

Mathematics, 22.06.2019 00:00

Aclothing designer is selecting models to walk the runway for her fashion show. the clothes she designed require each model’s height to be no more than y inches from 5 feet 10 inches, or 70 inches. which graph could be used to determine the possible variance levels that would result in an acceptable height, x?

Answers: 2

Answer

Mathematics, 22.06.2019 00:30

If you invest 1,500 today in a bank that gives you a 5 percent annual interest rate, which of these items can you buy in two years? a. electronics worth $1,650 b.fitness equipment worth $1,700 c.a holiday package worth $2,000

Answers: 2

Answer

You know the right answer?

\sqrt{53+20\sqrt{7}} can be written in the form a+b\sqrt{c}, where a, b, and c are integers and c ha...

Questions