Problem 62 - Algebra A Combined Function

Question

Use the quotient rule and simplify each expression. $$ \frac{9 a^{4} b^{7}}{27 a b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{3} a^{3} b^{5} \).

Step 1: Identify Numerator and Denominator

The given expression is \( \frac{9 a^{4} b^{7}}{27 a b^{2}} \). Identify the numerator as \( 9 a^{4} b^{7} \) and the denominator as \( 27 a b^{2} \).

Step 2: Simplify Coefficients

Divide the coefficients of the numerator and the denominator. The coefficient in the numerator is 9, and in the denominator, it is 27. Simplify \( \frac{9}{27} = \frac{1}{3} \).

Step 3: Apply the Quotient Rule for the Variable 'a'

Apply the quotient rule for the powers of \( a \). According to the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \), we have \( a^{4-1} = a^{3} \).

Step 4: Apply the Quotient Rule for the Variable 'b'

Apply the quotient rule for the powers of \( b \). Using the quotient rule, we get \( b^{7-2} = b^{5} \).

Step 5: Write the Simplified Expression

Combine the results from the above steps to construct the simplified expression. The expression becomes \( \frac{1}{3} a^{3} b^{5} \).

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, making them easier to understand and work with. It involves rewriting the expression in a way that reveals its basic structure or shows a simpler equivalent. In algebra, this process often involves condensing terms and applying mathematical rules, such as the quotient rule.

For example, consider the expression \( \frac{9 a^4 b^7}{27 a b^2} \). To simplify it, proceed by:

• Identifying common factors in both the numerator and the denominator.
• Reducing coefficients, i.e., simplifying \( \frac{9}{27} \) to \( \frac{1}{3} \).
• Applying algebraic rules to simplify the powers of variables.

Simplifying expressions helps in solving equations and inequalities more easily and also in performing algebraic operations like addition, subtraction, multiplication, and division.

Exponents

Exponents are a way of representing repeated multiplication of the same number by itself. In mathematics, an exponent indicates how many times the base number should be multiplied. For example, in the expression \( a^4 \), the base is \( a \) and the exponent is 4, meaning \( a \) is multiplied by itself four times.

When dealing with expressions that involve exponents, such as \( \frac{9 a^4 b^7}{27 a b^2} \), we often use the quotient rule for exponents. This rule states that when dividing like bases, you subtract the exponents, as shown here:

• For the variable \( a \), use \( a^{4-1} = a^3 \).
• For the variable \( b \), apply \( b^{7-2} = b^5 \).

This step is crucial for simplifying expressions involving powers, making complex algebraic tasks more manageable.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both consist of algebraic expressions. Simplifying algebraic fractions requires understanding how to manage expressions in a fraction form to make them look less complicated.

The given expression, \( \frac{9 a^4 b^7}{27 a b^2} \), is an example of an algebraic fraction. To simplify such fractions:

• You need to simplify both coefficients and variables separately.
• Use common mathematical rules, like the quotient rule for powers.
• Reduce coefficients to their simplest form, like \( \frac{9}{27} \) becoming \( \frac{1}{3} \).

Simplifying algebraic fractions is useful in many areas of algebra, allowing us to solve fractions more efficiently by reducing them step by step.