Problem 89 - Elementary Algebra

Question

Write the expression \(\frac{42 x^{2} y^{5} z^{3}}{21 x^{4} y^{7}}\) so that no denominator appears.

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{42x^{2}y^{5}z^{3}}{21x^{4}y^{7}}\). Answer: \(2z^{3} \cdot \frac{1}{x^{2}y^{2}}\)

Step 1: Cancel common factors in the numerator and denominator

First, we will find the greatest common divisor (GCD) for the numerical coefficients 42 and 21. Divide both by their GCD, which in this case is 21. \(\frac{42}{21} = 2\) Now look at the terms with variables. Cancel out the common exponents in the numerator and denominator by subtracting the smaller exponent from the larger exponent: \(x^{2}\) and \(x^{4}\): subtract exponents, \(4 - 2 = 2\) \(y^{5}\) and \(y^{7}\): subtract exponents, \(7 - 5 = 2\)

Step 2: Apply the quotient rule for exponents and simplify the expression

Since we cancelled out the common factors, we are left with: \(\frac{2x^{2-4}y^{5-7}z^{3}}{1}\) Now apply the quotient rule for exponents by subtracting the exponents: \(2x^{-2}y^{-2}z^{3}\) The expression can be rewritten without any denominators as follows: \(2z^{3} \cdot \frac{1}{x^{2}y^{2}}\)

Key Concepts

Simplification of Fractions

Simplifying fractions means making them as straightforward as possible while maintaining their value. This often involves canceling out common factors found both in the numerator and the denominator. By doing this, we lessen the complexity of fractions.

• Identify the Greatest Common Divisor (GCD): Check the numbers involved in the expression, and find the biggest number that divides them both. In our example, the numbers were 42 and 21. We determined their GCD to be 21. By dividing them, the fraction simplifies.
• Canceling Variables: If variables exist in a fraction, they can also be simplified by reducing coefficients or selecting common terms found in both the numerator and denominator. Subtract smaller exponents from larger ones to reduce the fraction's complexity without changing its value.

Simplifying fractions helps with easier calculations, clearer expressions, and better understanding of math problems.

Exponent Rules

Exponents are used when a number or variable is multiplied by itself a certain number of times. In algebraic fractions, understanding how to handle them correctly is crucial both for simplification and solving problems.

• Quotient Rule: This is a key exponent rule that helps in simplifying fractions. It states that when dividing similar bases, you subtract the exponents: \[a^m / a^n = a^{m-n}\]For example, in simplifying \(x^2/x^4\), we subtract: \(4 - 2 = 2\).
• Negative Exponents: Result from subtracting a bigger exponent at the bottom from a smaller one on top. They can be rewritten as a positive exponent by inverting their position—a negative exponent on the top can be moved to the bottom, and vice versa.

Being adept at handling exponents helps in rewriting expressions in a simpler, easier-to-understand form.

Factoring

Factoring is about breaking an expression down into simpler base components. It is a valuable skill when dealing with complex algebraic fractions since it can reveal factors that can be easily canceled out.

• Extracting Common Terms: This involves identifying terms that appear consistently across different parts of the expression. Removing these common factors can help simplify fractions drastically.
• Prime Factorization: Sometimes, for numerical expressions, breaking down numbers to their prime factors can help in quickly identifying which terms can be simplified or canceled.

Factoring is an essential skill for reducing any algebraic expression to its simplest form, ultimately easing the solving process and understanding of the algebraic concept.