how do you do the following without a calculator:
3 x 4^(1/4) - 12 x 2^(-3/2)
solve for the unknown:
6e^(-4t) = 2
ahh sorry that was supposed to be a separate question.
note that you can combine exponents if they have the same base. thus, we change the 4 (first term) to 2^2 since the base on the second term is equal to 2:
3 x 4^(1/4) - 12 x 2^(-3/2)
3 x (2^2)^(1/4) - 12 x 2^(-3/2)
also, since 12 = 2*2*3 = (2^2)*3,
3 x (2^2)^(1/4) - (3*2^2) x 2^(-3/2)
3 x 2^(1/2) - 3 x (2^2) x (2^(-3/2))
note that to multiply exponents, first they must have the same base, and then you can add their exponents. thus, for the second term, we just add the exponents of 2:
3 x 2^(1/2) - 3 x (2^2) x (2^(-3/2))
3 x 2^(1/2) - 3 x (2^(2 - 3/2))
3 x 2^(1/2) - 3 x 2^(1/2)
0
hope this helps~ :)
thank you! :]
To solve the expression 3 x 4^(1/4) - 12 x 2^(-3/2) without a calculator, we can break it down step by step using basic mathematical operations.
Let's start with the first term: 3 x 4^(1/4).
1. Simplify the exponent: 4^(1/4). The exponent 1/4 represents the fourth root of 4.
To calculate the fourth root of 4, we need to find a number that, when raised to the power of 4, equals 4. We can start with an estimate: 2^4 = 16, so the fourth root of 16 is 2. Since 4 is between 2^2 and 2^3, we can estimate that the fourth root of 4 is between 2 and 2.5.
To find a more precise value, we can use a method called "guess and check". We guess a number between 2 and 2.5, substitute it into the expression, and check if we get a value close to 4. By trying different values, we can approximate the fourth root of 4 to be approximately 1.189207.
Now, we simplify the expression: 3 x 1.189207.
Multiply 3 by the approximate value we found: 3 x 1.189207 = 3.567621.
Moving on to the second term: 12 x 2^(-3/2).
2. Simplify the exponent: 2^(-3/2). The negative sign in the exponent indicates that we need to take the reciprocal of the number to make the exponent positive, which gives us (1/2)^3.
To calculate (1/2)^3, we cube the reciprocal of 2, which is 1/2. (1/2)^3 = (1/2) x (1/2) x (1/2) = 1/8.
Now, we simplify the expression: 12 x 1/8 = 12/8 = 3/2 = 1.5.
Finally, we subtract the second term from the first term: 3.567621 - 1.5 = 2.067621.
Therefore, the solution to the expression 3 x 4^(1/4) - 12 x 2^(-3/2) without a calculator is approximately 2.067621.