Problem 113 - College Algebra Essentials

Question

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\frac{\left(2^{-1} x^{-2} y^{-1}\right)^{-2}\left(2 x^{-4} y^{3}\right)^{-2}\left(16 x^{-3} y^{3}\right)^{0}}{\left(2 x^{-3} y^{-5}\right)^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \( \frac{1}{x^{2}y^{18}} \).

Step 1: Simplify expressions inside parentheses

Begin by simplifying individual exponential expressions inside parentheses using the negative exponent rule, by which any quantity with a negative exponent in the numerator is moved to the denominator and vice versa, with its exponent sign switched. Likewise, simplify expressions with exponent 0 to 1: \[\left(\frac{1}{2^2} \frac{1}{x^4} \frac{1}{y^2}\right)\times\left(\frac{1}{4} \frac{1}{x^{-8}} \frac{1}{y^{-6}}\right)\times\left(16 \frac{1}{x^{0}} \frac{1}{y^{0}}\right)\div\left(4 \frac{1}{x^{-6}} \frac{1}{y^{-10}}\right)\]

Step 2: Group like terms together

Next, simplify the expression by grouping the constants, \(x\) terms, and \(y\) terms together across the multiply and divide operations: \[\frac{\left(\frac{1}{2^2}\times\frac{1}{4}\times 16\right)\times\left(\frac{1}{x^4}\times\frac{1}{x^{-8}}\times \frac{1}{x^{0}}\right)\times\left(\frac{1}{y^2}\times\frac{1}{y^{-6}}\times \frac{1}{y^{0}}\right)}{\left(4 \times \frac{1}{x^{-6}} \times \frac{1}{y^{-10}}\right)}\]

Step 3: Simplify the grouped terms

Now, perform the multiplication within each group and simplify. The rule of exponents states that when multiplying terms with the same base, the exponents should be added. Also, note that anything to the power of 0 equals 1: \[\frac{\left(\frac{16}{4}\right)\times\left(x^{8-4}\right)\times\left(y^{-6-2}\right)}{\left(4x^{6}y^{10}\right)}\]

Step 4: Simplify the final expression

Finally, simplify the final expression by performing the remaining arithmetic operations and subtraction in the exponentiation. The result will be the simplified version of the initial exponential expression:\[\frac{\left(4 \times x^{4} \times y^{-8}\right)}{\left(4x^{6}y^{10}\right)}\]

Step 5: Cancel out common factors

We can now cancel out common factors in the numerator and the denominator to simplify the fraction further. In this case, both the numerator and denominator have \(4x^{4}y^{-8}\) in common, so we end up with: \[\frac{1}{x^{2}y^{18}}\]