You can put this solution on YOUR website! I assume the equation is
and not
If so, please use parentheses on multiple term or fractional exponents. Clearly stated problems are more likely to get a response.
To solve
or any equation for that matter, it can be very helpful to keep in mid your goal. You goal is to get an equation of the form
x = something
or
something = x
Looking at the "x" in our goal we see that
It's coefficient is 1
li>Its exponent is 1
It is not part of some fraction. It is not in a denominator nor does it have a denominator (other than 1).
There are no other terms on that side of the equation.
None of this should be a surprise. But if we don't keep this in mind then we might miss some very simple solutions to problems.
Looking at
we should see that if we could find a way to change the exponent from 2/3 to a 1 then we would be done! So how do we change exponents? We have several rules for exponents. Most of them apply to multiplying or dividing x's with exponents. And if we were to use any of those rules, we would have to multiply or divide both sides by an x with an exponent. This would put an x on the left side, too. So those properties will not help us.
But there is one rule that does not involve multiply or dividing x's with exponents. And that is the power of a power rule: If we raise both sides of the equation to the same power then we will not get any x's on the left side. And if we raise both sides to the right power then we can change the exponent of 2/3 to a 1!
So what by what power should we raise both side of the equation to get an exponent of 1 on the x? The rule we are going to use tells us that when we raise a power to a power we should multiply the exponents. So what times 2/3 is equal to 1? If we're familiar with reciprocals we should know that one fact that is true about all reciprocals is that their product is always a 1! So what we want to do is raise each side of the equation to the reciprocal of 2/3 power. The reciprocal of 2/3 is 3/2. So we'll raise both sides to the 3/2 power:
The right side simplifies as we planned:
The only thing left is to simplify the left side of the equation. What is 4^(3/2)? If you're not sure comfortable with negative or fraction exponents then I find it can be helpful to factor the exponent is a special way:
If the exponent is negative, then factor out -1.
If the exponent is a fraction and the numerator is not not a 1, then factor out the numerator.
Factoring the exponent in this way:
1. factor out -1 if it is negative. Your exponent is not negative.
2. Factor out the numerator if it is not a 1:
With the exponent factored we can look at each individual factor and see what should be done to simplify. The factor of 3 in the exponent tells us that we need to cube something. And the factor of 1/3 in the exponent tells us that a square root should be done (since 1/2 as an exponent means square root). So we want to cube and to find a square root and ... it does not matter which we do first! so just choose the order that looks easiest. A square root of 4 looks easier to me than cubing 4 so I'll start with that:
(Note how I remove the relevant factor of the exponent after I've done operation it represented.)
And not I'll cube 2:
This may not have seemed to have been a quick solution. But if we take out all the explanations we get:
(