Answer:
There are 10 digits in 250¹¹ × 40¹⁹
Explanation:
To find the number of digits in 250¹¹×40¹⁹, we need to first calculate the value of this expression and then count the number of digits in the result.
To calculate the value of 250¹¹×40¹⁹, we can use the rule of exponents that says:
a¹ⁿ × b¹ᵐ = (a × b)¹ⁿᵐ
Using this rule, we can simplify the expression as:
250¹¹ × 40¹⁹ = (2⁵ × 5³)¹¹ × (2³ × 5)¹⁹
= 2¹¹ × 5³¹ × 2⁵ × 5¹⁹
= 2¹⁶ × 5⁵⁰
Now, we need to calculate the number of digits in 2¹⁶ × 5⁵⁰. To do this, we can take the logarithm of this number to the base 10, and then add 1 to the result. The integer part of the logarithm will give us the number of digits in the number.
So, we have:
log(2¹⁶ × 5⁵⁰) = log 2¹⁶ + log 5⁵⁰
Using the logarithm rule that says log aⁿ = n log a, we can simplify this expression as:
Using the logarithm rule that says log aⁿ = n log a, we can simplify this expression as:log(2¹⁶ × 5⁵⁰) = 16 log 2 + 50 log 5
Using the logarithm rule that says log aⁿ = n log a, we can simplify this expression as:log(2¹⁶ × 5⁵⁰) = 16 log 2 + 50 log 5Now, we need to evaluate log 2 and log 5. We can use a calculator to get:
log 2 ≈ 0.30103
log 5 ≈ 0.69897
Substituting these values, we get:
log(2¹⁶ × 5⁵⁰) ≈ 16 × 0.30103 + 50 × 0.69897
≈ 9.0321
Finally, adding 1 to this result, we get:
number of digits = ⌈log(2¹⁶ × 5⁵⁰)⌉ + 1
= ⌈9.0321⌉ + 1
= 10
Therefore, there are 10 digits in 250¹¹ × 40¹⁹