1/2^(-x)=3 then how is 4^(x+2)?
To solve the equation 1/2^(-x) = 3 for x, we can follow the steps below:
Step 1: Start by expressing 1/2^(-x) in a different form. The exponent -x in the denominator can be written as 2^x in the numerator. So, the equation becomes:
1/(1/2^x) = 3
Step 2: Simplify the equation by multiplying both sides by 2^x to get rid of the fraction:
1 * 2^x = 3
Step 3: Since any number raised to the power of 0 is equal to 1, we can rewrite the equation as:
2^x = 3
Step 4: To solve for x, we need to take the logarithm of both sides. In this case, we will use the logarithm with base 2 since we have 2^x:
log2(2^x) = log2(3)
Step 5: Since the logarithm of a power with the same base gives us the exponent, we can simplify further:
x = log2(3)
Now, to find 4^(x+2), we substitute the value of x we obtained from the previous equation:
4^(x+2) = 4^(log2(3) + 2)
To simplify this further, we need to use the properties of exponents. Specifically, we can use the property that (a^b) * (a^c) = a^(b+c):
4^(log2(3) + 2) = 4^(log2(3)) * 4^2
Since 4 can be written as 2^2, we can substitute and simplify:
4^(log2(3) + 2) = (2^2)^(log2(3)) * (2^2)
Using the exponent rule (a^b)^c = a^(b*c), we can simplify further:
4^(log2(3) + 2) = 2^(2 * log2(3)) * 4
Finally, using the property that 2^log2(x) = x, we can simplify the equation:
4^(log2(3) + 2) = 2^(2 * log2(3)) * 4 = 3 * 4 = 12
So, 4^(x+2) = 12 when x satisfies the equation 1/2^(-x) = 3.