Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left[\left(\frac{x^{2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}}\right)^{-2}\right]^{3} $$

Step-by-Step Solution

Answer

The simplified expression with positive exponents only is: \[ \frac{y^{12} z^{36}}{x^{24}} \]

Step 1: Analyze the given expression

Break down the given expression and identify the exponent rules to apply in simplifying. We start by analyzing the expression: \[ \left[\left(\frac{x^{2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}}\right)^{-2}\right]^{3} \] To simplify this expression, we would first simplify inner parentheses and then simplify outer ones. This means first simplifying the fraction, applying powers to powers rule inside the inner parentheses, and then apply the exponents outside the inner parentheses to get a simplified expression with only positive exponents.

Step 2: Simplify the fraction

Apply the quotient rule for exponents in the given fraction. The quotient rule for exponents states that a division operation can be simplified by subtracting the powers in the denominator from their respective powers in the numerator. \( \frac{x^{2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}} = x^{2-(-2)} y^{-3-(-1)} z^{-4-2} \)

Step 3: Add exponents

Simplify the expression by calculating the results of the exponent sums. Adding the exponents: \( x^{2-(-2)} y^{-3-(-1)} z^{-4-2} = x^{2+2} y^{-3+1} z^{-4-2} \) In simpler terms: \( x^4 y^{-2} z^{-6} \)

Step 4: Apply powers to powers rule

Simplify the expressions inside the parentheses by raising each term to the exponent of the parentheses. We will now apply the powers to powers rule: \( \left(\frac{x^{2} y^{-3} z^{-4}}{x^{-2} y^{-1} z^{2}}\right)^{-2} = \left(x^4 y^{-2} z^{-6}\right)^{-2} \) It simplifies to: \( x^{4\cdot(-2)} y^{-2\cdot(-2)} z^{-6\cdot(-2)} \)

Step 5: Multiply the exponents

Calculate the result of exponent multiplications. Multiplying the exponents: \( x^{4\cdot(-2)} y^{-2\cdot(-2)} z^{-6\cdot(-2)} = x^{-8} y^{4} z^{12} \) Finally, we have our simplified expression inside the inner parentheses: \( \left(x^{-8} y^{4} z^{12}\right) \)

Step 6: Apply cubed exponent

Simplify the expression further by raising the simplified term to the cubed exponent. Now, we apply the powers to powers rule again for the outer brackets: \( \left(\left(x^{-8} y^{4} z^{12}\right)\right)^{3} = x^{(-8) \cdot 3} y^{4\cdot 3} z^{12\cdot 3} \) This simplifies to: \( x^{-24} y^{12} z^{36} \)

Step 7: Convert to positive exponents

Rewrite the expression using positive exponents only. To convert negative exponents to positive: \( x^{-24} y^{12} z^{36} = \frac{y^{12} z^{36}}{x^{24}} \) Now the expression is written using positive exponents only. The simplified expression is: \[ \frac{y^{12} z^{36}}{x^{24}} \]

Key Concepts

Exponent Rules

Exponent rules are essential guidelines in simplifying expressions that involve powers. They make mathematical operations with exponents much easier to manage. Here's a quick overview of these rules:

• Product of Powers Rule: When multiplying two exponential terms with the same base, you add their exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• Quotient Rule: When dividing two exponential terms with the same base, subtract the exponent in the denominator from the exponent in the numerator (more on this below).
• Power of a Power Rule: When raising an exponent to another power, multiply the exponents together.
• Negative Exponent Rule: A negative exponent indicates a reciprocal. For example, \(a^{-m} = \frac{1}{a^m}\).

Understanding these rules allows for effective simplification of algebraic expressions, ensuring results are as simple and clear as possible.

Quotient Rule

The quotient rule is particularly useful in simplifying expressions that involve division of powers with the same base. The rule is straightforward:

When you divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

\[\frac{a^m}{a^n} = a^{m-n}\]
This rule helps to seamlessly handle expressions like \(\frac{x^2}{x^{-2}}\), simplifying it to \(x^{2-(-2)} = x^{4}\).
It is widely applied in various scientific and mathematical contexts. Learning to use the quotient rule effectively enhances your ability to work with complex algebraic expressions.

Powers of Powers

When dealing with nested exponents, such as raising a power to another power, the power of powers rule is your go-to method.

The rule says that to raise a power to another power, you multiply the exponents:
\[(a^m)^n = a^{m \cdot n}\]
For instance, if you have \( (x^4)^{-2} \), you apply the rule to get \( x^{4 \cdot (-2)} = x^{-8} \).
This powerful tool significantly simplifies expressions and is integral in advanced algebraic manipulations, making seemingly complex problems approachable.

Negative Exponents

Negative exponents can initially be confusing, but they simply indicate reciprocals or inverses.

Here's how to handle them:

• Basic Conversion: An expression like \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \).
• Application: For example, \( x^{-3} \) translates to \( \frac{1}{x^3} \).
• Simplification: When simplifying expressions, rewrite all terms with negative exponents as their reciprocals to express them as positive exponents only.

This change not only ensures consistency and clarity in your expressions but also makes further algebraic operations, such as multiplication or division, more straightforward. Embracing this rule can significantly enhance your problem-solving skills in algebra.