given that 9002.05 can be written as 9*10^3+2*10^a+5*10^b.are integers,find the value of a and b
help me
9*10^3 = 9000
So after 9002.05 is reduced by 9000 , we are left with
2.005 or
2 + .005
= 2*10^0 + 5*10^-3
so 9002.005
= 9*10^3+2*10^a+5*10^b
= 9*10^3 + 2*10^0 + 5.0*10^-3
a=0 , b = -3
Why did the number go to therapy?
Because it had too many exponents!
For the given number 9002.05, we can express it as a sum of powers of 10 in the form:
9 * 10^3 + 2 * 10^a + 5 * 10^b
To find the values of 'a' and 'b', we need to compare the powers of 10 in both the decimal and exponent parts.
From the given number, we can see that the decimal part has 2 digits after the decimal point. So, to maintain the decimal value of 0.05, we can express it as 5 * 10^(-2). Meanwhile, the exponent part should result in the remaining digits, 9002.
Comparing the powers of 10, we have:
10^3 = 10^(-2+a+b)
By equating the powers, we get:
3 = -2 + a + b
Simplifying the equation, we find:
a + b = 5
Therefore, the values of 'a' and 'b' that satisfy the equation are any pair of integers that add up to 5.
Keep in mind that the equation has multiple solutions, so there could be multiple valid values for 'a' and 'b'.
To find the values of a and b in the expression 9002.05 = 9 * 10^3 + 2 * 10^a + 5 * 10^b, we need to compare the powers of 10 in the given equation.
Let's start by comparing the powers of 10 in the units (ones) place:
On the left side of the equation, we have 0 in the units place, which means there is no power of 10 in the units place.
On the right side of the equation, we have 2 * 10^a + 5 * 10^b. To have 0 in the units place, both terms (2 * 10^a and 5 * 10^b) should have a 0 in the units place. This means that a must be greater than or equal to 1 since 10^1 = 10 and b must be greater than or equal to 1 since 5 * 10^b should have a 0 in the units place.
Now let's compare the powers of 10 in the thousands place:
On the left side of the equation, we have 10^3 (since 9002.05 is equal to 9 * 10^3 + 2 * 10^a + 5 * 10^b) which means there is a power of 10 equal to 3 in the thousands place.
On the right side of the equation, we do not have any power of 10 in the thousands place since both terms (2 * 10^a and 5 * 10^b) have a power of 10 smaller than 3.
Therefore, the value of a is less than 3 and the value of b is less than 3 as well.
To summarize:
- The value of a should be greater than or equal to 1 and less than 3.
- The value of b should be greater than or equal to 1 and less than 3.
To find the values of 'a' and 'b' in the expression 9*10^3 + 2*10^a + 5*10^b = 9002.05, we can compare the powers of 10 and the corresponding coefficients on both sides of the equation.
We know that 9002.05 can also be written as 9000 + 2 + 0.05.
Comparing the coefficients of the powers of 10 on both sides of the equation, we have:
- 9 * 10^3 on the left side vs. 9000 on the right side
- 2 * 10^a on the left side vs. 2 on the right side
- 5 * 10^b on the left side vs. 0.05 on the right side
From this comparison, we can deduce the values of 'a' and 'b' as follows:
- 'a' is the exponent of 10 that gives us 2. Since 10^1 = 10, 'a' must be equal to 1.
- 'b' is the exponent of 10 that gives us 0.05. We need to find the logarithm of 0.05 to the base 10. Using a scientific calculator, we find that log(0.05) = -1.30103. So, 'b' must be equal to -1.
Therefore, the values of 'a' and 'b' in the expression 9*10^3 + 2*10^a + 5*10^b = 9002.05 are a = 1 and b = -1.