Can you show me how to find (1.016)^3 without calculator?

Question

Anyone can do 1.016³ with a calculator but I am weak in decimal-related calculations and have to sit for exams that do not allow calculators. So can anyone show how to do this step by step without a calculator?

Answer 1

1.016)^3 = (1 + .016)^3

there's an algebraic formula: (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3)

the coefficients for the 4 terms are also found on the 4th row of Pascal's Triangle

1

11

121

1331

plug (1 + .016) values into the formula with a=1 and b = .016

(1)^3 + 3(1)^2(.016) + 3(1)(.016)^2 + (.016)3

= 1 + .048 + 3(.000256) + .000004096

= 1 + .048 + .000768 + .000004096

keep the decimal places in order when added

1

0.048

0.000768

0.000004096

=

1.048772096

one way to check this answer is the "rule of 9's"

add up the digits in the answer: 1+4+8+7+7+2+9+6 = 42, 4+2=6

then add up the digits in the problem 1.016: 1+1+6 = 8

8x8=64 6+4 =10 1+0=1 1 x 8 = 8

8=8. That doesn't guarantee the answer is correct. but it's a 90% chance it is.

or another method is

just do long division

1.016 times

1.016 equals

1.032256

1.016 times

1.016 equals

6096

1016

1016 add the columns

1.032256

now multiply that by 1.016

1.032256 times

1.016 to get

6193536

1032256

0

1032256

1.048772096 you can count the decimal places in the problem 6 in 1.032230 and 3 in 1.016 to get 6+3 = 9 n the answer 1.048772096 Or just know it's a number that's slightly more than 1, so the decimal place as to go right after the 1.

Answer 2

(1.016)^3

For a question like this, you multiply the base (1.016) 3 times (because the exponent is 3)

So you would have 1.016x1.016x1.016

Answer 3

You can use the binomial approximation (1+ε)ⁿ is approximately equal to 1+nε. This works for all n if ε<<1.

In your case (1.016)³ is about 1 + 3(.016) = 1.048 which can be done in your head.

Actual value: 1.04877