solve the equation by expressing each side as a power of the same base and then equating exponents
3125^x=1/square root 5
did you realize that 3125 is a power of 5, namely 3125 = 5^5
and √5 = 5^(1/2) then 1/√5 = 5^(-1/2)
so your equation becomes
(5^5)^x = 5^(-1/2)
5^5x = 5^(-1/2)
then 5x = -1/2
x = -1/10
To solve the equation 3125^x = 1/sqrt(5) using the method of expressing each side as a power of the same base, we need to find a common base for both sides.
Let's start by expressing both sides using the same base, which will be 5 since 3125 is equal to 5^5.
Rewriting the equation with 5 as the base, we have:
(5^5)^x = 1/sqrt(5)
Now, let's simplify the left side by applying the exponent rule, which states that when raising an exponent to another exponent, we multiply the exponents:
5^(5x) = 1/sqrt(5)
To simplify the right side, we can rationalize the denominator by multiplying both the numerator and denominator by sqrt(5):
5^(5x) = sqrt(5)/sqrt(5) * 1/sqrt(5)
5^(5x) = sqrt(5) / 5
Now we have both sides of the equation with the same base 5. Since the bases are the same, we can equate the exponents:
5x = 1/2
To solve for x, divide both sides of the equation by 5:
x = (1/2) / 5
x = 1/10
So the solution to the equation 3125^x = 1/sqrt(5) is x = 1/10.