Question

Simplify the following exponential expression. Assume that variables represent nonzero real numbers. ((xy⁻⁴)⁻⁴)/((x⁻⁴y)⁻⁵)

Answer 1

To simplify the exponential expression ((xy⁻⁴)⁻⁴)/((x⁻⁴y)⁻⁵), distribute the negative exponents, divide the expressions, and then apply the rules of exponents to get the simplified form y²¹/x²⁴.

Step-by-step explanation:

To simplify the given exponential expression ((xy⁻⁴)⁻⁴)/((x⁻⁴y)⁻⁵), we follow the rules of exponents regarding division and negative exponents. The Division of Exponentials involves dividing the digit term of the numerator by the digit term of the denominator and subtracting the exponents of the exponential terms. The rule for negative exponents, or 1 xn as per Equation A.9, denotes that we flip the construction to the denominator, meaning we perform a division rather than multiplication.

First, we need to distribute the negative exponents across the products inside the parentheses:

• (xy⁻⁴)⁻⁴ becomes x⁻⁴y⁴⁶
• (x⁻⁴y)⁻⁵ becomes x²⁰y⁻⁵

Now, substitute these expressions into the original equation:

x⁻⁴y¹⁶ / x²⁰y⁻⁵

Following that, we divide the exponents:

• For x: (-4) - (20) = -24, so we get x⁻²⁴
• For y: (16) - (-5) = 16 + 5 = 21, so we get y²¹

Combining these:

x⁻²⁴y²¹. According to the rule for negative exponents, we can now flip the x term to the denominator to yield:

y²¹/x²⁴.

This is our simplified exponential expression.