Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left(a^{2} b^{-3}\right)^{2}\left(a^{-2} b^{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression with positive exponents only is: \[ \frac{a^{10}}{b^{12}} \]

Step 1: Apply the power rule to each term in parentheses

Remember, the power rule states that \((a^m)^n = a^{mn}\). Apply this rule to each term in the parentheses. \[ \left(a^{2} b^{-3}\right)^{2}\left(a^{-2} b^{2}\right)^{-3} \] becomes \[ a^{2 \cdot 2} b^{-3 \cdot 2} a^{-2 \cdot (-3)} b^{2 \cdot (-3)} \]

Step 2: Simplify the exponents

Perform the multiplications of the exponents to simplify. \[ a^{4} b^{-6} a^{6} b^{-6} \]

Step 3: Apply the multiplication rule to combine terms

Recall that the multiplication rule states that \(a^m * a^n = a^{m+n}\). Apply this rule to combine the terms with the same base. \[ a^{4} a^{6} \cdot b^{-6} b^{-6} \] becomes \[ a^{4+6}b^{-6+(-6)} \]

Step 4: Simplify the exponents again

Perform the additions of the exponents to simplify. \[ a^{10}b^{-12} \]

Step 5: Write with positive exponents only

We can rewrite \(b^{-12}\) with a positive exponent by moving the term to the denominator. \[ a^{10} \frac{1}{b^{12}} \] The simplified expression with positive exponents only is: \[ \frac{a^{10}}{b^{12}} \]

Key Concepts

Exponent Laws

Exponent laws are essential for simplifying algebraic expressions involving powers. There are several rules to keep in mind:

• Power Rule: This rule states that when an exponentiated expression is raised to another power, the exponents multiply. Mathematically, it is given by \( (a^m)^n = a^{m \cdot n} \).


• Multiplication Rule: When multiplying two powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).


• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base. For example, \( a^{-n} = \frac{1}{a^n} \).

Understanding these rules allows you to transform expressions and simplify them effectively. Each rule plays a role in achieving the final form of the expression with positive exponents.

Simplification Process

Simplification involves breaking down an expression into its most basic form. The process reduces complexity and turns expressions easier to understand or evaluate. Let's see how to simplify the given expression, \( \left(a^{2} b^{-3}\right)^{2}\left(a^{-2} b^{2}\right)^{-3} \).The simplification follows these steps:

• Applying the Power Rule: Start by addressing the exponents inside the parentheses. Use the power rule to multiply the exponents. This results in \( a^4 b^{-6} a^6 b^{-6} \).


• Combining Like Terms: Use the multiplication rule to combine like terms. Add exponents of like bases: \( a^4 \cdot a^6 = a^{10} \) and \( b^{-6} \cdot b^{-6} = b^{-12} \).

This method systematically reduces the expression by using suitable exponent laws.

Positive Exponents

Working with positive exponents simplifies expressions and makes them easier to interpret. Breaking down negative exponents is crucial to getting expressions in a standard positive exponent form. The simplification of negative exponents is managed by repositioning the terms:

• A term with a negative exponent in the numerator moves to the denominator with a positive exponent.
• Conversely, a negative exponent in the denominator moves to the numerator.

For the expression \( a^{10}b^{-12} \), turning \( b^{-12} \) into a positive exponent means it becomes \( \frac{1}{b^{12}} \) in the denominator. Hence, the expression becomes \( \frac{a^{10}}{b^{12}} \), which is now in its positive exponent form. This provides a clearer and more usable expression, which simplifies understanding and further calculations.