Problem 113 - College Algebra

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 expression is \(\frac{x^{16}}{y^{16}}\).

Step 1: Address Negative Exponents

Rewriting all the negative exponents to their reciprocal forms, the expression becomes: \(\frac{\left(2^{2} x^{4} y^{2}\right) \left(2^{-2} x^{8} y^{-6} \right) \left(16^{0} x^{0} y^{0}\right)}{\left(2^{2} x^{6} y^{-10}\right)}\)

Step 2: Simplify the Zero Exponential Terms

Any number raised to the power of zero equals one. Therefore, the expression simplifies to: \(\frac{\left(2^{2} x^{4} y^{2}\right) \left(2^{-2} x^{8} y^{-6} \right)}{\left(2^{2} x^{6} y^{-10}\right)}\)

Step 3: Simplify the expression using multiplication and division rules of exponents

The full expression simplifies to: \(4x^{12}y^{-4} \times \frac{1}{4}x^{16}y^{-12} \div 4x^{12}y^{-20}\) which further simplifies to: \(x^{12}y^{-4} \times x^{16}y^{-12} \div x^{12}y^{-10}\) resulting to: \(x^{16}y^{-16} \)

Step 4: Convert Negative Exponents to Positive

The negative exponent rule states that we move the base with the negative exponent from the numerator to the denominator (or the from the denominator to the numerator) and change the sign of its exponent, thus the simplified expression is: \(\frac{x^{16}}{y^{16}}\)

Key Concepts

Negative Exponents

When encountering negative exponents, it's important to remember what they signify. A negative exponent indicates the reciprocal or inversely proportional form of the number or variable. For example, if you see \(x^{-n}\), it can be rewritten as \(\frac{1}{x^n}\). This is a key rule in manipulating and simplifying expressions involving exponents.

• In the provided exercise, the expression \(2^{-1}x^{-2}y^{-1}\) becomes \(\frac{1}{2} \times \frac{1}{x^2} \times \frac{1}{y} = \frac{1}{2x^2y}\).
• By switching negative exponents to their reciprocal form, it becomes easier to simplify the entire expression by multiplication or division.

This method makes expressions neater and is a fundamental step before applying other exponent rules.

Zero Exponent Rule

The zero exponent rule is straightforward: any base raised to the power of zero is equal to one. This is because there are no factors of the base left, essentially leaving just a value of 1.

• For instance, \(16^0\), \(x^0\), and \(y^0\) all simplify to 1.
• This rule greatly simplifies terms in an expression and can even eliminate them if they are being multiplied by others.

In the exercise, the component \(\left(16 x^{-3} y^{3}\right)^0\) simplifies to 1, effectively removing it from the equation—streamlining further computations.

Simplifying Expressions

Simplifying expressions is the process of making them as concise as possible. This often involves combining like terms, factoring out common factors, and utilizing rules for exponents.
In the exercise, once negative and zero exponents are addressed, you simplify using multiplication and division rules.

• Combine powers of the same base: \(a^m \times a^n = a^{m+n}\).
• Divide powers with the same base: \( \frac{a^m}{a^n} = a^{m-n}\).

After applying these rules, the exercise handles elements like \(x^{12}y^{-4} \) by adjusting exponents further to reach the most simplified final form.

Exponent Multiplication and Division Rules

The manipulation of exponential expressions relies heavily on multiplication and division rules. Mastering these rules can significantly ease the process of handling expressions with different bases and exponents.

• Multiplication of Same Base: When multiplying powers with the same base, you add their exponents: \(x^a \times x^b = x^{a+b}\).
• Division of Same Base: When dividing like bases, subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\).

In the exercise, these rules are crucial in reducing the complex expression to \(x^{16}\), after carefully managing each step with attention to multiplication and division nuances. Finally, ensuring that all exponents are positive solidifies the expression's simplicity and correctness.