Problem 21 - Algebra 1

Question

Simplify the expression. $$\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5}{6x} \).

Step 1: Rewrite Division as Multiplication

The expression can be rewritten as \(\frac{3 x^{2}}{10} * \frac{25}{9 x^{3}}\) by reciprocating the second fraction.

Step 2: Multiply Fractions

Perform multiplication between the fractions. Multiply the numerators together to form the new numerator and the denominators to form the new denominator getting \(\frac{75 x^{2}}{90 x^{3}}\).

Step 3: Simplify the Expression

Simplify the expression by cancelling out the common factors between the numerator and denominator. We get \(\frac{75 x^{2}}{90 x^{3}} = \frac{5}{6x}\) by cancelling \(15x^{2}\) out of the numerator and denominator and simplifying.

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is a crucial skill allowing you to transform complex equations into simpler forms, making them easier to solve or analyze.
The overall goal is to minimize the number of terms to its simplest version.

• Cancel Out Common Factors: Look for numbers or variables that appear in both the numerator and the denominator and eliminate them. This helps reduce the expression to a simpler equivalent.
• Constant Terms: Simplify numerical coefficients by finding common divisors. In other words, if both the numerator and the denominator have a common number, divide them by it.
• Variable Terms: Similarly, apply the same principle to the variable terms. Simplifying involves reducing powers of variables using the laws of exponents. Subtract the exponent of the denominator from the exponent of the numerator for the same base variable, as follows: if you have \(\frac{x^m}{x^n}\), simplify it to \(x^{m-n}\).

By regularly practicing simplifying expressions, solving algebra problems becomes much more manageable. This foundational skill often underpins solving equations and analyzing functions in various branches of mathematics.

Division of Fractions

Dividing fractions might initially seem complex, but it's a straightforward concept once you understand the technique known as "multiply by the reciprocal."

• Reciprocal Understanding: A reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
• Multiply by the Reciprocal: The division of fractions is redefined by multiplying the first fraction by the reciprocal of the second. It's often summarized in the rule: "Copy, Flip, Multiply." As in the exercise, \(\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}\) turns into \(\frac{3 x^{2}}{10} \cdot \frac{25}{9 x^{3}}\).
• Why Use Reciprocals? Multiplication is a more straightforward operation than division in fractions, making calculations easier through elimination and reduction.

Understanding the division of fractions is foundational for more complex algebraic equations, as it enables you to handle expressions efficiently.

Multiplying Fractions

When multiplying fractions, the process is simpler compared to addition or subtraction. Multiplication involves two key steps: multiply across numerators and multiply across denominators.

• Multiply Numerators: Take the top numbers (numerators) of each fraction and multiply them together. For instance, multiply \(3x^2\) by \(25\) to get \(75x^2\).
• Multiply Denominators: Similarly, multiply the bottom numbers (denominators) of each fraction. From the exercise, multiply \(10\) and \(9x^3\) yielding \(90x^3\).
• Fractions Simplification Opportunities: Before multiplying, check if you can simplify across numerators and denominators to reduce your expressions right from the start, which can save a lot of effort later on.

Multiplying fractions is an essential mathematical skill that enhances your ability to tackle both straightforward and more challenging problems in algebra. Mastery of this concept paves the way for success in further math studies including calculus and beyond.