Question

Find the value of 8 2/3-√9×10° + (1/144) -1/2

Answer 1

For 13^2 through 17^2, we have

13^2 = [13+3]9 =169; note 3^2 ends in 9

14^2 = [14+5]6=196; note 4^2 ends in 6

15^2=[15+7]5=225; note 5^2 ends in 5

16^2=[16+9]6=256; note 6^2 ends in 6

17^2=[17+11]9=289; note 7^2 ends in 9.

We just used the sequence {3, 5, 7, 11} to help us in this pattern that ends here.

For 18^2 and 19^2, you may have noticed that

18^2 = [4•8]4 = 324; note that 8^2 ends in 4; and

19^2 =[4•9]1= 361; note that 9^1 ends in 1.

Now, for a more general way at looking at the squares of whole numbers…

0^2 = 0 is given

1^2 = 0 + 0 + 1 = 1

2^2 = 1 + 1 + 2 = 4

3^2 = 4 + 2 + 3 = 9

4^2 = 9 + 3 + 4 = 16

5^2 = 16 + 4 + 5 = 25

18^2 = [4•8]4 = 324; note that 8^2 ends in 4; and

19^2 =[4•9]1= 361; note that 9^1 ends in 1.

Now, for a more general way at looking at the squares of whole numbers…

0^2 = 0 is given

1^2 = 0 + 0 + 1 = 1

2^2 = 1 + 1 + 2 = 4

3^2 = 4 + 2 + 3 = 9

4^2 = 9 + 3 + 4 = 16

5^2 = 16 + 4 + 5 = 25

7^2 = 36 + 6 + 7 = 49

8^2 = 49 + 7 + 8 = 64

9^2 = 64 + 8 + 9 = 81

10^2 = 81 + 9 + 10 = 100

11^2 = 100 + 10 + 11 = 121

12^2 = 121 + 11 + 12 = 144

13^2 = 144 + 12 + 13 = 169

14^2 = 169 + 13 + 14 = 196

15^2 = 196 + 14 + 13 = 225

16^2 = 225 + 15 + 16 = 256

17^2 = 256 + 16 + 17 = 289

18^2 = 289 + 17 + 18 = 324

19^2 = 324 + 18 + 19 = 361

20^2 = 361 + 19 + 20 = 400, etc.

Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

${8}^{ \frac{2}{3} } - \sqrt{9} ( {10)}^{0} + \bigg( \frac{1}{144} \bigg) ^{ - \frac{1}{2} }$

$= ( {8}^{2} {)}^{ \frac{1}{3} } - 3(1) + \frac{1}{ \bigg( \frac{1}{144} \bigg) ^{ \frac{1}{2} } }$

$= \sqrt[3]{64} - 3 + \frac{1}{ \left \{ \bigg( \frac{1}{12} \bigg) ^{2} \right \}^{ \frac{1}{2} } }$

$= \sqrt[3]{4 \times 4 \times 4} - 3 + \frac{1}{\bigg( \frac{1}{12} \bigg)}$

$= 4 - 3 + 12$

$= \boxed{13 \: \bf{Ans.}}$