Jacob estimates the square root of 75 in the following way:

√75=3√25=3 * 5=15

(a) Explain why his reasoning is incorrect.

(b) Estimate the square root of 75 to the nearest thousandths without using your calculator. Show your work.

Note: If you use a calculator you will get the wrong answer.

a)

Step-by-step explanation:

Square root of 75 (√75) is regarded as a surd function because the value inside the square root sign is not a perfect square. In that case the given value will be splitted into the product of a largest perfect square and a prime number as shown:

√75 = √3*25

√75 = √3*√25

√75 = √3 * 5

√75 = 5√3

a) Based on the above calculation, it can be inferred that his reasoning is incorrect. The value of 3 was supposed to be inside a square root not a whole number i. e √3 * √25

b) Since √75 = 5√3

√75 = 5 * 1.7321

√75 = 8.6605

√75 = 8.661 {to nearest thousand}

Answer: √3 = 1.732

Step-by-step explanation:

Solution:-

- Jacob tries to estimate the square root of 75 and makes a calculation error. The following work was done by Jacob:

√75 = 3√25 = 3*5 = 15

- The error was made by Jacob when he pulled out the constant " 3 " from under the root and missed to apply the property of square root on " 3 " itself.

- The correct version would be:

√75 = √3*√25 = √3*5

Answer: 5√3

- Now we will estimate the square root of 3 without using a calculator:

- We will first two per squares above and below " 3 ".

[ √4 = 2, √1 = 1 ]

- From above we know that the square root lies within the range of [ 1, 2 ].

- We also know that " √3 " would is closer to "√4 ". Hence, would be closer to 2 as compared to 1.

- We will take a guess of " 1.75 ", and square all guesses:

Guess 1: 1.75^2 = 3.0625

- The number is higher than "3" but very close. Guess a number lower than last one but closer to 1.75:

Guess 2: 1.73^2 = 2.9929

- The number is lower than "3" but very close. Guess a number higher than last one but closer to 1.73:

Guess 3: 1.735^2 = 3.010225

- The number is higher than "3" but very close. Guess a number lower than last one but closer to 1.735:

Guess 4: 1.7325^2 = 3.00155625

- The number is higher than "3" but very close. Guess a number lower than last one but closer to 1.7325:

Guess 4: 1.7323^2 = 3.00086329 ... Stop here

- Rounding the number to nearest thousandths we need to look at the 4th decimal place value of " 1.7323 ". The 4th decimal place has to be lower than 3 to get an exact √3.

- Hence, the answer to nearest thousandth would have all the 4th decimal places an onwards truncated. Therefore,

Answer: √3 = 1.732