Problem 104 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(\frac{3 a^{4}}{9 b^{5}}\right)^{2} $$

Step-by-Step Solution

Answer

\( \frac{a^8}{9b^{10}} \)

Step 1: Distribute the Exponent

First, distribute the exponent 2 across the fraction and each part inside the parentheses. Use the property \((\frac{a}{b})^n = \frac{a^n}{b^n}\). So, we have: \[\left(\frac{3a^4}{9b^5}\right)^2 = \frac{(3a^4)^2}{(9b^5)^2}\]

Step 2: Apply the Power to Each Factor

Next, apply the exponent to each factor in the numerator and the denominator separately: \[(3a^4)^2 = 3^2 \cdot (a^4)^2 = 9a^8\] \[(9b^5)^2 = 9^2 \cdot (b^5)^2 = 81b^{10}\] This gives: \[\frac{9a^8}{81b^{10}}\]

Step 3: Simplify the Fraction

Finally, simplify the fraction by dividing the coefficients and applying the properties of exponents. The simplified form of the fraction is: \[\frac{9a^8}{81b^{10}} = \frac{9}{81} \cdot \frac{a^8}{b^{10}} = \frac{1}{9} \cdot \frac{a^8}{b^{10}}\] Therefore, the simplified expression is: \[\frac{a^8}{9b^{10}}\]

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill that makes complex algebraic expressions more manageable. In this process, both the numerator and the denominator are divided by their greatest common factor (GCF). The goal is to reduce the fraction to its simplest form, where no further division is possible.
For example, in the expression \(\frac{9}{81}\), both the numerator and the denominator can be divided by 9, which is their GCF. This simplifies to \(\frac{1}{9}\).

• Step-by-step division: Begin by identifying the largest number that divides both the top and bottom.
• Use the GCF to reduce both sides evenly.

Whenever you simplify a fraction involving variables, look for any common factors among their powers as well. This method not only applies to numbers but extends to algebraic terms where you can apply the properties of exponents.

Properties of Exponents

Understanding the properties of exponents is key when dealing with expressions such as \( (\frac{3a^4}{9b^5})^2 \). These properties help break down complex powers into simpler terms.
Here are some vital properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\). When multiplying, you add the exponents if the bases are the same.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\). Multiply the exponents when raising a power to another power.
• Power of a Fraction: \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Distribute the exponent to both the numerator and the denominator.

These properties allow you to transform and simplify expressions, making the mathematics more straightforward. Knowing how to apply these rules will enable you to manipulate expressions for simplification or expansion.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In expressions such as \( \left(\frac{3a^4}{9b^5}\right)^2 \), variables like \(a\) and \(b\) hold any value, making the expression general.
Understanding how to work with algebraic expressions involves mastering operations like addition, subtraction, multiplication, and division while using rules like order of operations (PEMDAS).

• Combining Like Terms: Terms that have the same variable raised to the same power can be combined.
• Distributive Property: This is essential for expanding expressions. For example, \(a(b+c) = ab + ac\).

Applying these concepts to simplify or rearrange algebraic expressions will aid in solving equations and understanding functions. Mastery of these topics is essential for advancing in algebra and beyond.