Problem 70 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{2 a^{-1} b}{a^{2} b^{-3}}\right)^{-3} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{a^9}{8 b^6} \).

Step 1: Apply Negative Exponent Rule

Recall that any number raised to a negative exponent can be expressed as the reciprocal of that number with a positive exponent: \[ x^{-n} = \frac{1}{x^n} \].Apply this rule to the entire expression,\[ \left( \frac{2 a^{-1} b}{a^{2} b^{-3}} \right)^{-3} = \left( \frac{a^{2} b^{-3}}{2 a^{-1} b} \right)^{3} \].

Step 2: Simplify Inside the Parentheses

Simplify the expression inside the parentheses: \[ \frac{a^{2} b^{-3}}{2 a^{-1} b} \].For the variables:- Base \(a\): \( a^{2} \times a = a^{2+1} = a^3 \)- Base \(b\): \( b^{-3} \times b^{-1} = b^{-3+1} = b^{-2} \)So it becomes: \[ \frac{a^3}{2 b^2} \].

Step 3: Distribute the Power of 3

Apply the exponent 3 to each component of \( \left( \frac{a^3}{2 b^2} \right)^3 \):\[ (a^3)^3 = a^{3 \times 3} = a^9 \]\[ (b^2)^3 = b^{2 \times 3} = b^6 \]\[ (2)^3 = 8 \] This yields: \[ \frac{a^9}{8 b^6} \].

Step 4: Final Simplified Expression

Combine all parts into a single expression: \[ \frac{a^9}{8 b^6} \], which is the simplified form with all exponents positive.

Key Concepts

Negative Exponents

When we talk about negative exponents, we're dealing with a special rule for handling these mathematical expressions. A negative exponent essentially tells us to "flip" the base to the other side of a fraction. For example, if we have the expression \( x^{-n} \), it implies that \( x \) is on the bottom of a fraction with a positive exponent: \( x^{-n} = \frac{1}{x^n} \).
This rule is incredibly useful for simplifying complex expressions, especially those involving fractions.

• If the base with the negative exponent is in the numerator, it moves to the denominator as a positive exponent.
• Conversely, if it resides in the denominator, it moves to the numerator with a positive exponent.

In our exercise, we applied the negative exponent rule to transform the entire fractional expression. This step is crucial as it sets the stage for further simplification.

Exponent Rules

Exponent rules, or laws, provide us with a set of guidelines for performing operations on exponential expressions easily and efficiently. These rules include several key concepts:

• The product of powers rule: \( x^a \times x^b = x^{a+b} \).
• The power of a power rule: \( (x^a)^b = x^{a \times b} \).
• The power of a product rule: \( (xy)^a = x^a \times y^a \).

Utilizing these rules allows us to manipulate the powers of numbers and variables systematically. In the given problem, the second step involved using these rules to simplify terms inside the parentheses. For instance, by applying power rules, we combined like bases, adjusted their exponents, and made them easier to handle. This step is essential as it reduces complicated expressions into manageable parts for further calculations and simplifications.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra which involves using rules and properties to make expressions as concise and simple as possible. By doing so, we can easily interpret or solve mathematical equations.
The ultimate goal of simplification is to express the problem or equation in its most reduced form. In the context of exponent expressions, this involves:

• Canceling out similar terms.
• Converting any negative exponents to positive using the negative exponent rule.
• Simplifying fractional expressions by reducing to the most basic forms of each component.

In our exercise, after applying the relevant exponent rules and converting negative exponents, we were left with an expression \( \frac{a^9}{8 b^6} \). This is the most simplified version, with all exponents positive and the expression easy to evaluate or manipulate further if necessary. Simplifying expressions helps in making complex mathematical problems easier to solve and comprehend by removing unnecessary clutter.