QuestionCalculate the value of $\sqrt{459}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the square root of459. The square root of a number is a value that, when multiplied by itself, gives the original number.

STEP 2

The square root of a number can be found using a calculator. However, if we want to solve it manually, we can use the long division method or the approximation method. Here, we will use the approximation method.

STEP 3

First, we need to find two perfect squares that459 lies between. The perfect square smaller than459 is $20^2 =400$ and the perfect square larger than459 is $21^2 =441$ .

STEP 4

Since459 is closer to441, we can say that the square root of459 is slightly more than21.

STEP 5

To get a more accurate result, we can use the formula for approximating a square root $\sqrt{a} \approx x + \frac{a - x^2}{2x}$ where $a$ is the number we are taking the square root of and $x$ is the estimate we have.

STEP 6

Plug in the values for $a$ and $x$ into the formula.
$\sqrt{459} \approx21 + \frac{459 -21^2}{2 \cdot21}$

STEP 7

Calculate the value inside the fraction.
$\sqrt{459} \approx21 + \frac{459 -441}{42}$

STEP 8

Calculate the value of the fraction.
$\sqrt{459} \approx21 + \frac{18}{42}$

STEP 9

implify the fraction.
$\sqrt{459} \approx21 + \frac{3}{7}$

STEP 10

Calculate the final approximation of the square root.
$\sqrt{459} \approx21.43$ The square root of459 is approximately21.43.

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QuestionCalculate the value of $\sqrt{459}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the square root of459. The square root of a number is a value that, when multiplied by itself, gives the original number.

STEP 2

The square root of a number can be found using a calculator. However, if we want to solve it manually, we can use the long division method or the approximation method. Here, we will use the approximation method.

STEP 3

First, we need to find two perfect squares that459 lies between. The perfect square smaller than459 is $20^2 =400$ and the perfect square larger than459 is $21^2 =441$ .

STEP 4

Since459 is closer to441, we can say that the square root of459 is slightly more than21.

STEP 5

To get a more accurate result, we can use the formula for approximating a square root $\sqrt{a} \approx x + \frac{a - x^2}{2x}$ where $a$ is the number we are taking the square root of and $x$ is the estimate we have.

STEP 6

Plug in the values for $a$ and $x$ into the formula.
$\sqrt{459} \approx21 + \frac{459 -21^2}{2 \cdot21}$

STEP 7

Calculate the value inside the fraction.
$\sqrt{459} \approx21 + \frac{459 -441}{42}$

STEP 8

Calculate the value of the fraction.
$\sqrt{459} \approx21 + \frac{18}{42}$

STEP 9

implify the fraction.
$\sqrt{459} \approx21 + \frac{3}{7}$

STEP 10

Calculate the final approximation of the square root.
$\sqrt{459} \approx21.43$ The square root of459 is approximately21.43.

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QuestionCalculate the value of 459\sqrt{459}459​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the square root of459. The square root of a number is a value that, when multiplied by itself, gives the original number.

STEP 2

The square root of a number can be found using a calculator. However, if we want to solve it manually, we can use the long division method or the approximation method. Here, we will use the approximation method.

STEP 3

First, we need to find two perfect squares that459 lies between. The perfect square smaller than459 is 202=40020^2 =400202=400 and the perfect square larger than459 is 212=44121^2 =441212=441.

STEP 4

Since459 is closer to441, we can say that the square root of459 is slightly more than21.

STEP 5

To get a more accurate result, we can use the formula for approximating a square roota≈x+a−x22x\sqrt{a} \approx x + \frac{a - x^2}{2x}a​≈x+2xa−x2​where aaa is the number we are taking the square root of and xxx is the estimate we have.

STEP 6

Plug in the values for aaa and xxx into the formula.459≈21+459−2122⋅21\sqrt{459} \approx21 + \frac{459 -21^2}{2 \cdot21}459​≈21+2⋅21459−212​

STEP 7

Calculate the value inside the fraction.459≈21+459−44142\sqrt{459} \approx21 + \frac{459 -441}{42}459​≈21+42459−441​

STEP 8

Calculate the value of the fraction.459≈21+1842\sqrt{459} \approx21 + \frac{18}{42}459​≈21+4218​

STEP 9

implify the fraction.459≈21+37\sqrt{459} \approx21 + \frac{3}{7}459​≈21+73​

STEP 10

Calculate the final approximation of the square root.459≈21.43\sqrt{459} \approx21.43459​≈21.43The square root of459 is approximately21.43.

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QuestionCalculate the value of $\sqrt{459}$ .

First, we need to find two perfect squares that459 lies between. The perfect square smaller than459 is $20^2 =400$ and the perfect square larger than459 is $21^2 =441$ .

To get a more accurate result, we can use the formula for approximating a square root $\sqrt{a} \approx x + \frac{a - x^2}{2x}$ where $a$ is the number we are taking the square root of and $x$ is the estimate we have.

Plug in the values for $a$ and $x$ into the formula.
$\sqrt{459} \approx21 + \frac{459 -21^2}{2 \cdot21}$

Calculate the value inside the fraction.
$\sqrt{459} \approx21 + \frac{459 -441}{42}$

Calculate the value of the fraction.
$\sqrt{459} \approx21 + \frac{18}{42}$

implify the fraction.
$\sqrt{459} \approx21 + \frac{3}{7}$

Calculate the final approximation of the square root.
$\sqrt{459} \approx21.43$ The square root of459 is approximately21.43.