Math Grade 8, fractional indices
Hi, i am having trouble with the following questions from the lesson fractional indices, please answer with all steps. thanks.
Find the value of x:
1) 4^x = 64
2) x^2 = 4/9
Please answer them based on the method i have learnt, which is like this: the following is an example, and i would like to learn it the same way.
16^1/4 = 4 sqrt16 = 2
You need to solve for x the equation `4^x = 64` , hence, since `64 = 4*4*4 = 4^3` , you may substitute `4^3` for 64 such that:
`4^x =4^3 `
Since the left and right sides are equal, hence, you need to equate the exponents such that:
`x = 3`
Hence, evaluating the solution to the equation `4^x = 64 ` yields`x = 3` .
You need to solve for x the equation `x^2 = 4/9` , hence, you need to raise both sides to the power `1/2` such that:
`(x^2)^(1/2) = (4/9)^(1/2) `
Using the exponents properties and converting the rational power into a radical yields:
`(x)^(2*1/2) = +-sqrt(4/9) => x_(1,2) = +-2/3`
Hence, evaluating the solutions to the equation `x^2 = 4/9` yields `x_(1,2) = +-2/3` .
4^x = 64
taking x sqrt both sides
x sqrt(4^x)= x sqrt(64)
when only x=3, left hand side and right hand side become similar, so the answer is x=3
similarly, x^2=4/9 can be solve
2 sqrt (x^2)=2 sqrt (4/9)
so the answer is 2/3