Question

Instructional Item 2

Use the properties of exponents to create an equivalent expression for the given expression
shown below with no variables in the denominator.
(64x²)-^-1/6(32x^5)^-2/5

Answer 1

Recall that the following properties of exponents hold for nonzero real numbers a and b and integers m and n:

a^m * a^n = a^(m+n)
(a^m)^n = a^(mn)
(a*b)^m = a^m * b^m
a^(-m) = 1/a^m

Using these properties, we can simplify the given expression as follows:

(64x²)^(-1/6) * (32x^5)^(-2/5)
= (2^6 * x^2)^(-1/6) * (2^5 * 2x^5)^(-2/5)
= 2^(-6/6) * x^(-2/6) * 2^(-2/5) * (2x^5)^(-2/5)
= 2^(-1) * x^(-1/3) * 2^(-2/5) * 2^(-4/5) * x^(-2)
= 2^(-9/5) * x^(-7/3)
= (1/2^(9/5)) * (1/x^(7/3))

Thus, an equivalent expression with no variables in the denominator is (1/2^(9/5)) * (1/x^(7/3)).

Answer 2

The equivalent expression with no variables in the denominator is 1/(2x^2).