leiana_chung
Write (8a^-3)^-2/3 in simplest form.
So far I have this.
= (8^-2/3)(a^3(-2/3))
=(8^-2/3)(a^2)
=((2^3)^-2/3)(a^2)
=(2^-2)(a^2)
And this is where I am stuck, can someone explain how to do the rest of the steps?
actually Nickie; you are on the right path; if you multiply the (1/4) with the (a^2), you'll get a^2/4, which is the right answer
Note that since 8 = 2^3,
8a^-3 = 2^3/a^3 = (2/a)^3
so, (8a^-3)^-2/3 = (a/2)^2 = a^2/4
You are done, and you are correct.
Thanks, Steve! Always knew I count on YOU, Steve! *wheezes*
I think the next step is this
=(1/2^2)(a^2)
=(1/4)(a^2)
But I do need help with this now.
To simplify the expression (8a^(-3))^(-2/3), let's break it down step by step.
Step 1: Simplify the base.
We have (8a^(-3))^(-2/3).
The base 8 can be simplified as 2^3, so we can rewrite our expression as ((2^3)a^(-3))^(-2/3).
Step 2: Apply the power rule.
According to the power rule, when a power is raised to another power, we multiply the exponents.
In this case, we have ((2^3)a^(-3))^(-2/3), so we multiply the exponents: 3 × -2/3 and -3 × -2/3.
This gives us ((2^3)a^(-6/3))^(-2/3) = (8a^(-2))^(-2/3).
Step 3: Invert the fraction exponent.
To invert the fraction exponent, we switch the numerator and denominator.
So, (8a^(-2))^(-2/3) becomes 1/(8a^(-2))^(2/3).
Step 4: Apply the power rule again.
Now that we have 1/(8a^(-2))^(2/3), we can apply the power rule by multiplying the exponents.
This gives us 1/(8^(2/3)a^((-2)(2/3))).
Simplifying further:
1/(8^(2/3)a^(-4/3)).
Step 5: Express 8^(2/3) in simplest form.
To simplify 8^(2/3), we can rewrite it as (2^3)^(2/3).
Using the power rule, (2^3)^(2/3) becomes 2^(3 × 2/3) = 2^2 = 4.
So, our expression becomes 1/(4a^(-4/3)).
Step 6: Simplify the negative exponent.
Recall that a negative exponent can be rewritten as a positive exponent by moving it to the denominator.
Therefore, a^(-4/3) transforms into 1/a^(4/3).
Putting it all together, we have:
1/(4a^(4/3)).
And that is the simplified form of (8a^(-3))^(-2/3).