Question

Estimate the value of each expression to the nearest integer and to the nearest tenth

Answer 1

We have to find the square root of the no. & multiply it with -1 to find the answer. Also, the answer should be to the nearest integer & the nearest 10th. Using division method:

A) - √10

• Write 10 as 10.00.. Pair the digits. Start from the decimal & move away from it. Hence, 1st pair would be 10, 2nd would be 00, & 3rd would be 00 and so on
• Start with the 1st pair, & find the no. whose square is ≤ first pair. So here, since first pair is 10, the no. whose square is just ≤ 10 is 3. Write 3 in the quotient and subtract 3² from 10 to get a remainder of 1
• We have reached the decimal in the dividend, so add a decimal in the quotient. Bring down the next pair to join with the remainder. The resulting no. is 100. Double the quotient (so 3 becomes 6) & use it to determine the divisor for 100. The divisor will be of the form 6x, & will be such that when it is multiplied with its unit digit then the product is ≤ 100 (i.e. 6x * x ≤ 100). So for x = 1, 61 *1 = 61, which is < 100. Append 1 to the quotient. The quotient now is 3.1. And subtract 31 (31*1) from 100 to find the remainder. The remainder is 39
• Bring down the next pair to join with the remainder. The resulting no. is 3900. Double the quotient (so 31 becomes 62) & use it to determine the divisor for 3900. The divisor will be of the form 62x, & will be such that when it is multiplied with its unit digit then the product is ≤ 3900 (i.e. 62x * x ≤ 3900). So for x = 6, 626 *6 = 3756. Append this 6 in the quotient. The quotient now is 3.16. And subtract 3756 from 3900 to find the remainder. The remainder is 144. We have square root of 10 up-till the 100th, so the answer is

- √10 = (-1) × √10 = (-1) × (3.16...) = -3.16..

Nearest 10th: - √10 = -3.2

Nearest integer: - √10 = -3

B) - √245

• Pair the digits. Start from the decimal & move away from it. Hence, 1st pair would be 2, 2nd would be 45, & 3rd would be 00 & so on
• Start with the 1st pair, & find the no. whose square is ≤ the 1st pair. So that no. whose square is ≤ 2 is 1. Write 1 in the quotient and subtract 1² from 2 to get a remainder of 1.
• Bring down the next pair. The resulting no. is 145. Double the quotient (so 1 becomes 2) & use it to determine the divisor for 145. The divisor will be of the form 2x, & will be such that when it is multiplied with its unit digit then the product is ≤ 145. So for x = 5, 25 *5 = 125, which is < 145. Append the 5 in the quotient. The quotient is 15. And subtract 125 (15*5) from 145 to find the remainder. The remainder is 20
• We have reached the decimal in the dividend, so add a decimal in the quotient. Bring down the next pair. The resulting no. is 2000. Double the quotient (so 15 becomes 30) & use it to determine the divisor for 2000. The divisor will be of the form 30x, & will be such that when it is multiplied with its unit digit then the product is ≤ 2000. So for x = 6, 306 *6 = 1836. Append the 6 to the quotient. The quotient now is 15.6. And subtract 1836 from 2000 to find the remainder. The remainder is 164.
• Bring down the next pair. The resulting no. is 16400. Double the quotient (so 156 becomes 312) & use it to determine the divisor for 16400. The divisor will be of the form 312x, & will be such that when it is multiplied with its unit digit then the product is ≤ 16400. So for x = 5, 3125 *5 = 62604. Append this 5 in the quotient. The quotient now is 15.65. We have square root of 245 up-till the 100th, so the answer is

- √245 = (-1) × √245 = (-1) × (15.65...) = -15.65....

Nearest 10th: - √245 = -15.7

Nearest integer: - √245 = -16

C) - √52

• Pair the digits. Start from the decimal & move away from it. Hence, 1st pair would be 52, 2nd would be 00, & so on
• Start with the 1st pair, & find the no. whose square is ≤ the 1st pair. So no. whose square is ≤ 52 is 7. Write 7 in the quotient & subtract 7² from 52 to get a remainder of 3
• We have reached the decimal in the dividend, so add a decimal in the quotient. Bring down the next pair, i.e. 00. The resulting no. is 300. Double the quotient (so 7 becomes 14) & use it to determine the divisor for 300. The divisor will be of the form 14x, & will be such that when it is multiplied with its unit digit the product is ≤ 300. So for x = 2, 142*2 = 284, which is < 300. Append the 2 in the quotient. The quotient now is 7.2. And subtract 284 (142*2) from 300 to find the remainder. The remainder is 16
• Bring down the next pair. The resulting no. is 1600. Double the quotient (so 72 becomes 144) & use it to determine the divisor for 1600. The divisor will be of the form 144x, & will be such that when it is multiplied with its unit digit then the product is ≤ 1600. So for x = 1, 1441*1 = 1441. Append the 1 to the quotient. The quotient now is 7.21. And subtract 1441 from 1600 to find the remainder. The remainder is 159. We have square root of 52 up-till the 100th, so the answer is

- √52 = (-1) × √52 = (-1) × (7.21...) = -7.21....

Nearest 10th: - √52 = -7.2

Nearest integer: - √52 = -7

Answer 2

$3^{2} = 9$ and $4^{2} = 16$

As 9 is the nearest integer to 10, then best approximation for $-\sqrt{10}$ is - 3.

$15^{2} = 225$ and $16^{2} = 256$

As 256 is the nearest integer to 245, then best approximation for $-\sqrt{245}$ is - 16.

$7^{2} = 49$ and $8^{2} = 64$

As 49 is the nearest integer to 52, then best approximation for $-\sqrt{49}$ is - 7.