Problem 76 - Algebra 2 and Trigonometry

Question

In \(74-82,\) write each expression as a power with positive exponents in simplest form. $$ \left(\frac{4 a^{4} b^{6}}{25 a^{-1} b}\right)^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

\( \frac{2}{5} a^{2.5} b^{2.5} \)

Step 1: Simplify the Expression Inside the Parentheses

Given the expression \( \left(\frac{4 a^{4} b^{6}}{25 a^{-1} b}\right)^{\frac{1}{2}} \), begin by simplifying the fraction inside the parentheses.Apply the law of exponents for division: \( \frac{a^m}{a^n} = a^{m-n} \). For \( a \):\[ a^{4 - (-1)} = a^{4 + 1} = a^5 \] For \( b \):\[ b^{6-1} = b^5 \] The expression now becomes \( \left(\frac{4 a^{5} b^{5}}{25}\right)^{\frac{1}{2}} \).

Step 2: Separate the Fraction for Each Base

The expression can be written as \( \left(\frac{4}{25} \cdot a^5 \cdot b^5 \right)^{\frac{1}{2}} \).This allows us to take the square root of each part individually, which simplifies further steps.

Step 3: Simplify the Numerical Fraction

Take the square root of the fraction component \( \frac{4}{25} \):\[ \sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5} \]

Step 4: Apply the Exponent Power Rule

Use the exponent rule \( (x^m)^n = x^{m \cdot n} \).Apply this to \( a^5 \) and \( b^5 \):\[ (a^5)^{\frac{1}{2}} = a^{5 \cdot \frac{1}{2}} = a^{2.5} \] \[ (b^5)^{\frac{1}{2}} = b^{5 \cdot \frac{1}{2}} = b^{2.5} \]

Step 5: Write the Expression with Positive Exponents in Simplest Form

Combine all parts to write the expression fully simplified:\[ \left(\frac{4 a^{4} b^{6}}{25 a^{-1} b}\right)^{\frac{1}{2}} = \frac{2}{5} \cdot a^{2.5} \cdot b^{2.5} \]

Key Concepts

Laws of Exponents

Understanding the laws of exponents is critical to simplifying expressions, especially when dealing with complex fractions involving variables. Exponents represent repeated multiplication. The basic laws help simplify expressions by reducing higher powers and complex fractions.

• **Product of Powers:** When multiplying like bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• **Quotient of Powers:** When dividing like bases, subtract one exponent from the other: \( \frac{a^m}{a^n} = a^{m-n} \).
• **Power of a Power:** When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• **Negative Exponent Rule:** A negative exponent indicates reciprocal: \( a^{-n} = \frac{1}{a^n} \).

Using these rules allows for the simplification of expressions into more manageable forms, making calculations easier and reducing the potential for errors. In our example, by rewriting and applying these exponents rules, we efficiently manage to simplify the expression ensuring all exponents remain positive.

Fraction Simplification

Fraction simplification is all about making expressions easier to work with by reducing them to their simplest form. In mathematics, fractions are simplified by finding any common factors in the numerator and the denominator and dividing them through.When variables with exponents are involved, remember:

• **Simplify Numerators and Denominators Separately:** Handle any numbers and variables separately to ensure correct simplification.
• **Apply the Laws of Exponents:** Use laws such as the Quotient of Powers to simplify terms within a fraction, like \( \frac{a^m}{a^n} = a^{m-n} \).

In the given problem, simplifying the fraction inside the parentheses required applying the laws of exponents to both \( a \) and \( b \) to eliminate negative exponents. As shown in the solution, by removing these, the entire fraction is vastly simplified and set up for further simplification with square roots.

Expression with Positive Exponents

Achieving an expression with only positive exponents is often the goal in simplification problems. Positive exponents make equations clearer and more standardized for interpretation and manipulation.To consistently arrive at positive exponents, consider these strategies:

• **Convert Exponents:** Use the rule \( a^{-n} = \frac{1}{a^n} \) to convert negative exponents to their reciprocal forms.
• **Simplify Step by Step:** Break down each component of the expression using the laws of exponents. For example, \( a^{4 - (-1)} = a^5 \) converts an initially negative exponent situation into a positive one.
• **Consolidate Terms:** Whenever possible, consolidate terms using other laws such as Product or Quotient of Powers to achieve these positive results across the equation.

In our exercise, efficiently using these techniques helped turn \( a^{-1} \) into a positive exponent by adjusting with a simple mathematical operation. The simplification process concluded with terms like \( a^{2.5} \) and \( b^{2.5} \), ensuring clarity and consistency throughout the expression.