We want to solve the given equations.
The first equation is x raised
to the power of 3/5 equals 27.
The first step will be to eliminate the fractional exponent
by raising both sides of the equation
to the power of the denominator.
Remember, when we have powers raised to powers,
we multiply the exponents, and because 3/5 times five
is equal to three, if we raise both sides
of the equation to the fifth power,
we can eliminate the fractional exponent.
So, on the left side, again, because we have powers
raised to powers, we multiply the exponents.
If it's helpful, we can write five as five over one.
Before multiplying, notice how the fives simplify out,
leaving us with just three times one, or three, which now
we have x cubed is equal to 27 raised to the fifth power,
and now from here, to solve for x,
we need to undo the cubing, and we can do this two ways.
Because three times 1/3 is equal to one,
we could raise both sides of the equation
to the 1/3 power, or we can also
undo the cubing by taking the cube root
of both sides of the equation.
So, let's write this equation down here below,
and we'll take a look at solving it two different ways.
Again, when am I using rational exponents
is to raise both sides of the equation
to the power of the reciprocal of three,
which is 1/3, so I'll raise both sides
of the equation to the 1/3 power.
Because we have powers of powers, we multiply the exponents.
Three times 1/3 is 3/3 or one, giving us x to the first,
or just x equals, on the right side, we have 27
to the fifth, raised to the 1/3 power.
Multiplying the exponents, we have 27
to the 5/3 power, and now let's go
to the calculator and evaluate this.
We have 27 raised to the power of five divided
by three, enter, which gives us 243.
x equals 243, or going back to the equation
where we had x cubed equals 27 to the fifth,
we can also solve for x by undoing the cubing
by taking the cube root of both sides of the equation.
So, let's also show this.
The cube root undoes the cubing.
The left side simplifies to x.
We have x equals the cube root of 27 to the fifth,
which, going back to the calculator,
if we press math and then option four,
we have the cubed root, and then
we have 27 raised to the fifth, enter,
and of course we get the same result.
Let's look at a second example.
Here, we have x to the 5/7 equals 243.
Again, for the first step, we'll raise
both sides of the equation to the power
of the denominator to eliminate
the fractional irrational exponent.
So, because the denominator is seven,
we'll raise both sides of the equation to the seventh power.
Multiplying the exponents, notice
how the sevens simplify out, leaving us with x to the fifth.
We have x to the fifth equals 243
raised to the seventh power, and now,
again, at this step, we have two choices.
We can solve for x using a rational
exponent, or using a radical.
So, let's go ahead and write this equation again
down here below, and we'll look at both methods.
Using rational exponents, we would raise
both sides of the equation to the power
of the reciprocal of five, or five over one, which is 1/5.
So, let's raise both sides of the equation to the 1/5 power.
On the left, multiplying the exponents,
five times 1/5 is 5/5, or one.
x to the first is just x.
We have x equals 243.
Here, we multiply the exponents,
so we have 243 raised to the 7/5 power.
Let's evaluate this on the calculator.
243 raised to the power of 7/5, or seven
divided by five, which gives us 2,187.
Or, using radicals, to undo the fifth power,
we can take the fifth root of both sides of the equation.
The fifth root undoes the fifth power,
giving us just x on the left.
x equals the fifth root of 243 raised to the seventh.
Going back to the calculator, we press math,
and then because we have the fifth root,
we select option five, we enter the index,
which is five, right arrow, and then we enter 243
raised to the seventh under the fifth root,
enter, and of course we get the same result.
I hope you found this helpful.