Write (8a^-3)^-2/3 in simplest form
1/4a^2
Please help me understand how you got to that answer.
apply the exponent of -2/3 to each of the factors
(8a^-3)^(-2/3)
= (8^(-2/3) )(a^2)
= (1/8^(2/3))(a^2)
= (1/4)a^2
To simplify the expression (8a^(-3))^(-2/3), we can start by simplifying the exponent -2/3.
The rule for negative exponents is that a^(-n) is equal to 1/a^n. Applying this rule, we rewrite a^(-3) as 1/a^3.
Now, let's plug this back into our original expression: (8(1/a^3))^(-2/3).
To simplify further, we can distribute the exponent of -2/3 to both the 8 and the fraction containing a:
8^(-2/3) * (1/a^3)^(-2/3).
For the base 8, we need to rewrite it with a rational exponent. The rule states that x^(m/n) is equal to the n-th root of x to the power of m.
Therefore, we can write 8^(-2/3) as the cube root of 8 raised to the power of -2. Simplifying this, we get:
(8^(1/3))^(-2) * (1/a^3)^(-2/3).
The cube root of 8 is equal to 2, so we can rewrite the first part as 2^(-2).
Next, let's simplify the fraction (1/a^3)^(-2/3).
Using the rule for negative exponents, we can rewrite (1/a^3)^(-2/3) as (a^3/1)^(2/3).
This becomes a^(6/3), which simplifies to a^2.
Now, our expression is reduced to: 2^(-2) * a^2.
Finally, we can simplify 2^(-2). Since any number to the power of -2 is equal to 1 divided by that number squared, we have:
1/2^2 * a^2.
Simplifying further, we get:
1/4 * a^2.
Therefore, the simplified form of (8a^(-3))^(-2/3) is 1/4 * a^2.