Problem 18 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \left(\frac{a^{6}}{b^{9}}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified expression of \( \left(\frac{a^{6}}{b^{9}}\right)^{5} \) is \( \frac{a^{30}}{b^{45}} \).

Step 1: Identify the base and the exponent

In the given expression \( \left(\frac{a^{6}}{b^{9}}\right)^{5} \), \( a^{6} \) and \( b^{9} \) are base terms and 5 is the exponent.

Step 2: Apply the power of a power property

This property states that \( (a^{n})^{m} = a^{nm} \). In this problem, apply this rule to each term of the quotient separately. This leads to \( a^{6*5} \) and \( b^{9*5} \).

Step 3: Simplify the expression

Perform the multiplication to simplify the exponent, which results in \( a^{30} \) and \( b^{45} \). So the simplified form of the expression is \( \frac{a^{30}}{b^{45}} \).

Key Concepts

Power of a Power

In mathematics, understanding the 'Power of a Power' property is very useful for simplifying expressions. This property helps you work with exponents, especially when exponents themselves are raised to further powers.
For instance, consider an expression such as \( (x^m)^n \). According to the 'Power of a Power' rule, you multiply the exponents: \( x^{m \times n} \). This means, starting with a base \( x \) raised to one power \( m \) and then raising it to another power \( n \), you can find the solution by multiplying the two exponents.

• This principle allows simplifying complex expressions into simpler ones.
• It reduces steps in solving equations and problems involving multiple levels of exponents.

This property is vital in algebra and helps build a foundation for more advanced topics in mathematics.

Base and Exponent

The terms 'Base' and 'Exponent' are fundamental when dealing with powers and exponents. Understanding these terms lays the groundwork for mastering the simplification of expressions.
In the expression \( a^n \), the 'base' is \( a \) and the 'exponent' is \( n \). The base represents the number that is being multiplied by itself, while the exponent tells us how many times to use the base in the multiplication.

• For example, \( 2^3 \) means the base \( 2 \) is used in multiplication three times: \( 2 \times 2 \times 2 \).
• The base remains the same, and the exponent determines the number of factors in the expression.

Recognizing these components of an expression simplifies the process of applying various properties of exponents, such as the 'Power of a Power' property, discussed earlier.

Simplifying Expressions

'Simplifying Expressions' is a crucial skill in algebra that involves reducing complex algebraic expressions into simpler forms. It helps in finding quick and easy solutions to mathematical equations.
To simplify an expression like \( \left(\frac{a^6}{b^9}\right)^5 \), you need to systematically apply the properties of exponents.
First, identify each part of the expression and recognize opportunities to apply rules such as the 'Power of a Power'. Here, by raising \( a^6 \) and \( b^9 \) to the 5th power, multiplying the exponents results in \( a^{30} \) and \( b^{45} \).

• Step-by-step reduction of the expression makes the process logical and manageable.
• Simplification transforms a complex algebraic fraction into an easier-to-handle form: \( \frac{a^{30}}{b^{45}} \).

Developing this skill enhances problem-solving abilities and mathematical fluency, making it a cornerstone of algebraic operations.