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11111nata11111 [884]
2 years ago
12

What is the least number which is to be added or subtracted to 651201 to get a perfect square?

Mathematics
1 answer:
Elina [12.6K]2 years ago
8 0

Answer:

Add 48 to get a perfect square

Explanation:

A perfect square would be of the number 651249 which means that you have to add 48 to get a perfect square.

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If x = (√2 + 1)^-1/3 then the value of x^3 + 1/x^3 is​
Shtirlitz [24]

Step-by-step explanation:

<u>Given</u><u>:</u> x = {√(2) + 1}^(-1/3)

<u>Asked</u><u>:</u> x³+(1/x³) = ?

<u>Solution</u><u>:</u>

We have, x = {√(2) + 1}^(-1/3)

⇛x = [1/{√(2) + 1}^(1/3)]

[since, (a⁻ⁿ = 1/aⁿ)]

Cubing on both sides, then

⇛(x)³ = [1{/√(2) + 1}^(1/3)]³

⇛(x)³ = [(1)³/{√(2) + 1}^(1/3 *3)]

⇛(x)³ = [(1)³/{√(2) + 1}^(1*3/3)]

⇛(x)³ = [(1)³/{√(2) + 1}^(3/3)]

⇛(x * x * x) = [(1*1*1)/{√(2) + 1)^1]

⇛x³ = [1/{√(2) + 1}]

Here, we see that on RHS, the denominator is √(2)+1. We know that the rationalising factor of √(a)+b = √(a)-b. Therefore, the rationalising factor of √(2)+1 = √(2) - 1. On rationalising the denominator them

⇛x³ = [1/{√(2) + 1}] * [{√(2) - 1}/{√(2) - 1}]

⇛x³ = [1{√(2) + 1}/{√(2) + 1}{√(2) - 1}]

Multiply the numerator with number outside of the bracket with numbers on the bracket.

⇛x³ = [{√(2) + 1}/{√(2) + 1}{√(2) - 1}]

Now, Comparing the denominator on RHS with (a+b)(a-b), we get

  • a = √2
  • b = 1

Using identity (a+b)(a-b) = a² - b², we get

⇛x³ = [{√(2) - 1}/{√(2)² - (1)²}]

⇛x³ = [{√(2) - 1}/{√(2*2) - (1*1)}]

⇛x³ = [{√(2) - 1}/(2-1)]

⇛x³ = [{√(2) - 1}/1]

Therefore, x³ = √(2) - 1 → → →Eqn(1)

Now, 1/x³ = [1/{√(2) - 1]

⇛1/x³ = [1/{√(2) - 1] * [{√(2) + 1}/{√(2) + 1}]

⇛1/x³ = [1{√(2) + 1}/{√(2) - 1}{√(2) + 1}]

⇛1/x³ = {√(2) + 1}/[{√(2) - 1}{√(2) + 1}]

⇛1/x³ = [{√(2) + 1}/{√(2)² - (1)²}]

⇛1/x³ = [{√(2) + 1}/{√(2*2) - (1*1)}]

⇛1/x³ = [{√(2) + 1}/(2-1)]

⇛1/x³ = [{√(2) + 1}/1]

Therefore, 1/x³ = √(2) + 1 → → →Eqn(2)

On adding equation (1) and equation (2), we get

x³ + (1/x³) = √(2) -1 + √(2) + 1

Cancel out -1 and 1 on RHS.

⇛x³ + (1/x³) = √(2) + √(2)

⇛x³ + (1/x³) = 2

Therefore, x³ + (1/x³) = 2

<u>Answer</u><u>:</u> Hence, the required value of x³ + (1/x³) is 2.

Please let me know if you have any other questions.

3 0
2 years ago
Martin deposits $200 in a savings account that earns 5% annual interest. four years later, cary deposits $200 in an account earn
Scrat [10]
<span>Martin deposits $200 in a savings account that earns 5% annual interest.

year         interest                       balance

1              200 * 5%                    200(1.05)

2              200(1.05) * 5%          200(1.05)^2

3              200(1.05)^2*5%        200(1.05)^3

y                                                200(1.05)^y

=> m = 200 (1.05)^y

four years later, cary deposits $200 in an account earning the same interest.

</span>
<span><span>year         interest                       balance

5              200 * 5%                    200(1.05)

6              200(1.05) * 5%          200(1.05)^2

7             200(1.05)^2*5%        200(1.05)^3

y                                                200(1.05)^(y-4)

=> c = 200(1.05)^ (y-4)
</span>
Answer:
Martin: 200(1.05)^y
Cary: 200(1.05)^(y–4)
</span>
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24 dived by 7=what is the remainder
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3 is the remainder because the closest is 7 times 21 so it’s remainder 3
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