Problem 74 - Algebra 2 and Trigonometry

Question

In \(74-82,\) write each expression as a power with positive exponents in simplest form. $$ \left(\frac{2 a^{\frac{1}{2}}}{3 a^{6}}\right)^{6} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{64}{729a^{33}} \) with positive exponents.

Step 1: Distribute the Exponent

Distribute the exponent of 6 to both the numerator and the denominator. Apply the rule \((\frac{a}{b})^n = \frac{a^n}{b^n}\) to rewrite the entire expression: \[\left(\frac{2 a^{\frac{1}{2}}}{3 a^{6}}\right)^{6} = \frac{(2 a^{\frac{1}{2}})^{6}}{(3 a^6)^{6}}\].

Step 2: Simplify the Numerator

Simplify \((2 a^{\frac{1}{2}})^{6}\) by applying the power of a product rule \((ab)^n = a^n b^n\). So, \((2 a^{\frac{1}{2}})^{6} = 2^6 (a^{\frac{1}{2}})^{6}\equiv 2^6 a^{6 \cdot \frac{1}{2}}\equiv 64 a^{3}\).

Step 3: Simplify the Denominator

Simplify \((3 a^6)^{6}\) by applying the power of a product rule again: \((3 a^6)^{6} = 3^{6}(a^{6})^{6} = 3^{6} a^{36}\).

Step 4: Simplify the Fraction

Combine the results from Step 2 and Step 3 into a single fraction: \[\frac{64 a^3}{729 a^{36}}\].

Step 5: Simplify the Exponents

Use the properties of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the expression further. So, \(\frac{a^3}{a^{36}} = a^{3-36} = a^{-33}\).

Step 6: Final Simplified Form

Use the result from Step 5 to express the fraction with positive exponents: \[\frac{64}{729} \cdot \frac{1}{a^{33}}\equiv \frac{64}{729a^{33}}\].

Key Concepts

Properties of Exponents

The properties of exponents provide us with powerful rules to simplify expressions where numbers are raised to powers. These rules are essential when working with algebraic expressions, making calculations more straightforward. Here are some important properties to keep in mind:

• Product of Powers: When multiplying two exponents with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Quotient of Powers: When dividing two exponents with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent: Any non-zero number raised to the zero power is 1: \(a^0 = 1\).
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the opposite positive power: \(a^{-n} = \frac{1}{a^n}\).

Applying these properties helps us transform complicated expressions into simpler ones. It is crucial to practice using these rules to gain confidence and solve problems more efficiently.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both are algebraic expressions, often involving variables and exponents. Simplifying these requires a good understanding of not only arithmetic with fractions but also the properties of exponents.When simplifying algebraic fractions, there are key steps to follow:

• Simplify numerators and denominators separately using exponent rules.
• Use the rule \(\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n\) to split and simplify complex fractions.
• Look for common factors in the numerator and denominator that can be canceled.
• Reduce expressions to their simplest form by applying the properties of exponents.

Understanding these concepts allows for effective manipulation of algebraic fractions, transforming them to a simpler, more manageable form. Being familiar with the properties of exponents is especially useful here, as they often guide us through simplifying such expressions.

Power of a Product Rule

The power of a product rule is a crucial tool when dealing with expressions where multiple factors are raised to an exponent. It states that when you have a product raised to a power, you can distribute the exponent to each factor individually. Formally, the rule is written as:\[(ab)^n = a^n b^n\]Here's how it applies:

• When given an expression such as \((2a)^{6}\), apply the exponent separately to each factor: \(2^6 \times a^6\).
• This separation allows each factor to be simplified on its own, making calculations easier.
• In algebraic expressions, this rule is often employed to simplify the numerator and the denominator before performing additional operations, such as dividing or subtracting exponents.

The power of a product rule streamlines the process of working with complex expressions. By breaking down components, it helps clarify each part's role in the larger calculation. Remembering and practicing this rule simplifies solving a broad range of algebra problems, especially those involving multiple variables and exponents.