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PT and Its Converse: Estimating Square Roots

1. To Estimate a square root, you must first list all of your perfect squares here up to
100:
2. To Estimate a cube root, you must first list all of your perfect cubes here up to
1000:

Next, find the perfect square or perfect cube that your number lies between.

Someone please help, I will give brainliest!

Respuesta :

Answer:

1) 10^2

2) 100^2

Step-by-step explanation:

I believe this is the right answer...

let me know of it helps.

:)

The estimate of the square roots and cube roots of numbers can be found by an iterative process;

The Pythagoras's Theorem, PT, and its Converse: Estimating Square Roots

  • The steps to estimating square roots are;

1. The perfect squares that are close to the number for which the square root is sought are listed

Example, to find the square root of 5, we have;

2², 5, 6, 7, 8, 3²

Therefore, the perfect square close to 5 are those of 2 and 3

The number 5 is then divided by the root of the number with closest perfect square, which is √4 = 2

Therefore, we have;

  • [tex]\dfrac{5}{2} = 2.5[/tex]

The average of the result of the division and the divisor is then found as follows;

  • [tex]\dfrac{2 + 2.5}{2} = 2.25[/tex]

The required number is then divided again by the average, 2.25, again, to get;

  • [tex]\dfrac{5}{2.25} = 2.\overline 2[/tex]

The average of the new quotient and the divisor is found again as follows;

  • [tex]\dfrac{2.25 + 2.\overline 2}{2} = 2.36\overline 1[/tex]

Repeat dividing the required number by the average again as follows;

  • [tex]\dfrac{5}{2.36 \overline 1} = 2.23602484472[/tex]

The average is found again as 2.23606797792

Dividing 5 gives 2.23606797708

Therefore;

  • 2.236067978 is a good estimate of √5 approximation

2. To estimate a cube root of a number, the closest, numbers with a perfect cube are listed

To estimate the cube root of 9, the closest perfect cubes are 8 and 27

We have;

∛8 < ∛9 < ∛27

2 < ∛9 < 3

Given that we have;

The difference between 9 and 8 = 1

The difference between 9 and 27 = 18

Therefore, the cube root of 9, ∛9 ≈ 2

For more accurate value, we have

[tex]{} \hspace {8pt}2.08\\3| \overline {9.000 000}\\ {} \hspace {-3pt} -8\\ {} \hspace {8pt} 1 \ 000 \ 0[/tex]

Where, the 2 in the quotient is given from the closest perfect cube to 9;

2³ = 8 < 9

The 8 is given from 2³ = 8

The 1 is given from 9 - 8 = 1

The 000, is obtained by bringing down 000, from the dividend

Next; Multiply 300 by 2² to get 1,200

1,200 is larger than 1,000, which is the remainder, therefore, a decimal point and 0 is added to the quotient, and a zero is added to the remainder to give 10,000, from which we have;

[tex]\dfrac{10,000}{300 \times 2^2} = 8\frac{1}{3}[/tex]

The quotient, 8 is added to the quotient line after the zero, to get;

∛9 ≈ 2.08

Learn more here:

https://brainly.com/question/24981685

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