Problem 1 - Algebra 2

Question

Simplify each expression. \(\frac{45 m n^{3}}{20 n^{7}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{9m}{4n^{4}} \).

Step 1: Identify the Given Expression

We are given the expression to simplify: \( \frac{45 m n^{3}}{20 n^{7}} \). This expression consists of coefficients (numbers) and variables (letters with exponents). First, we identify the coefficients and variables in the numerator and the denominator.

Step 2: Simplify the Coefficients

Simplify the coefficients 45 and 20 by finding their greatest common divisor (GCD). The GCD of 45 and 20 is 5. Divide both the numerator and the denominator by 5: \( \frac{45}{20} = \frac{9}{4} \). Now, this simplifies the expression to \( \frac{9m n^{3}}{4 n^{7}} \).

Step 3: Simplify the Variables

We need to simplify \( n^3 \) over \( n^7 \) using the law of exponents. Recall that when dividing powers with the same base, you subtract the exponents: \( n^{3} - n^{7} = n^{(3-7)} = n^{-4} \). Substitute back into the expression to get \( \frac{9m}{4n^{4}} \).

Step 4: Write the Final Simplified Expression

After simplifying both the coefficients and the variables, the simplified expression is \( \frac{9m}{4n^{4}} \).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key concept in simplifying algebraic expressions. It refers to the largest number that divides two numbers without leaving a remainder. For example, in the exercise above, the numbers 45 and 20 were part of the expression that needed simplification. To simplify these numbers, you find the GCD. Breaking each number down into its prime factors can reveal the largest divisor they share.

• 45 can be factorized as: 3 × 3 × 5
• 20 can be factorized as: 2 × 2 × 5

The common factor between these two numbers is 5, making it the GCD. When you divide both 45 and 20 by 5, you simplify the fraction from 45/20 to 9/4. This step reduces the complexity of the expression and is essential in making algebraic expressions more manageable, especially when combined with variables.

Law of Exponents

The law of exponents is a set of rules that govern how to simplify expressions involving powers. One common rule is that when you divide terms with the same base, you subtract the exponents. This is crucial in simplifying algebraic expressions with similar variables and different powers.In the exercise, the variable part of the expression had powers of 'n' in both the numerator and the denominator:

• Numerator: n^3
• Denominator: n^7

According to the law of exponents, you subtract the exponent of the denominator from the exponent of the numerator: \[ n^{3 - 7} = n^{-4} \] This simplified expression highlights how the law of exponents helps reduce complex expressions, particularly when dealing with similar terms spread across a fraction. Always remember, subtract the lower from the higher and keep the base constant.

Dividing Powers with the Same Base

Dividing powers with the same base is a practical method used in algebra that simplifies expressions by reducing exponents. This technique is part of the larger set of rules known as the laws of exponents. It involves reducing complexity by performing subtraction of exponents when bases are identical.In our problem, we encountered dividing the powers of \( n \) from the numerator and denominator (\( n^3 \) and \( n^7 \), respectively). Here's how it works:- When you divide \( n^3 \) by \( n^7 \), according to the law, you subtract the exponents: \[ n^{3} / n^{7} = n^{3-7} = n^{-4} \]A negative exponent indicates a reciprocal, making \( n^{-4} = \frac{1}{n^4} \). Implementing this simplifies the algebraic fraction further.This method ensures clarity and simplicity in expressions and is integral in transforming cumbersome algebraic phrases into something that’s easier to interpret and use.