Simplify the expression
(a^-5/4*c^1/4)^3
To simplify the expression (a^(-5/4) * c^(1/4))^3, we can apply the power of a product rule, which states that (ab)^n = a^n * b^n. Using this rule, we can simplify the expression as follows:
(a^(-5/4))^3 * (c^(1/4))^3
= a^(-15/4) * c^(3/4)
To further simplify this expression, we can use the power of a power rule, which states that (a^n)^m = a^(n*m). Applying this rule, we get:
a^(-15/4) * c^(3/4)
= (a^(-15/4) * c^0) * c^(3/4)
= a^(-15) * c^(3/4)
Therefore, the simplified expression is a^(-15) * c^(3/4).
To simplify the expression (a^(-5/4) * c^(1/4))^3, we can apply the exponent rule.
First, let's simplify the exponents inside the parentheses:
a^(-5/4) becomes 1/a^(5/4) (since a^(-n) = 1/a^n)
c^(1/4) remains the same.
Now, let's simplify the expression inside the parentheses:
(1/a^(5/4) * c^(1/4))^3 = 1^3 / (a^(5/4))^3 * (c^(1/4))^3
The numerator remains 1 since any number raised to the power of 0 is 1.
For the denominator, we can apply the power rule:
(a^(5/4))^3 = a^((5/4)*3) = a^(15/4)
Similarly, for the second term in the denominator:
(c^(1/4))^3 = c^((1/4)*3) = c^(3/4)
Finally, putting it all together, we have:
1 / (a^(15/4) * c^(3/4))
Therefore, the simplified expression is 1 / (a^(15/4) * c^(3/4)).