How do we solve this radical equation: (x + 1) ^ (3/2) = 64

Raise each side to the inverse of the exponent on the left side. This results in the exponent on the left side becoming 1,

(x + 1) ^ (3/2) = 64
((x + 1) ^ (3/2))^2/3 = 64^2/3 the left side then becomes just (x + 1), and the right side become 16
x + 1 = 16
x = 16 - 1 = 15
check this answer: (x + 1 = 16
16^3/2 = 4096.
sqrt(4096) = 64 Great!!

Remember the golden rule of algebra, what you do to one side, you must do to the other. Lets try to get that polynomial on the left by itself (without a power). Now don't confuse not having a power, with having a power of zero, that's not the case. Any number to the zero power is actually 1. What you want on the left is to have (x+1)^1 which is equal to x+1. If the left hand side is raised to the 3/2 power, what do you want to do to it to make it go to the first power. The answer is simple, raise it to the 2/3 power. Do you remember how powers work. (x^n)^m = x^(n*m). When you raise a quantity by a power, and then that whole quantity is raised to another power (in your case 2/3), you multiply the exponents. In your case, you wanted the power to be 1 so you can isolate x by itself through simple subtraction, so something raised to the 3/2 which is then raised again to the 2/3 will cause the power to go to 1, as 3/2 * 2/3 = 1. What you do to one side, you do to the other, so take 64 to the 2/3 power. In other words, square 64 and then take the cubed root of that; or if you really wanted to you could first take the cubed root of 64 and then square your result, both methods will give you the same answer. In any event you come to the conclusion x+1 = 16, in other words, x=15.

(x+1)^(3/2)=64
(x+1)=64^(2/3)
Note that 64 is 8^2, and 8=2^3.
(x+1)=(2^6)^(2/3)=2^(6*2/3)=2^(2*2)=2^…

x=16-1
x=15

You could also use a calculator to evaluate 64^(2/3). Logs will probably not be as helpful because it is harder to think about logs base (3/2)

(x+1)^(3/2)=64
(x+1)^((3/2)*(2/3))=64^(2/3)
x=(64^(2/3))-1
x= use your calculator pls.

(x + 1) ^ (3/2) = 64

( (x + 1) ^ (3/2) ) ^ (2/3) = 64^(2/3) <==== In this step we raise both sides to the 2/3 power.

x+1 = 16

x=15 [ANSWER]

Power both sides by 2/3:

x + 1 = 16

Subtract 1 from both sides:

x = 15