Problem 23 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ \frac{1}{8} x^{4} y^{3}, \quad \frac{1}{2} x y^{3} $$

Step-by-Step Solution

Answer

Answer: The other factor is $\frac{1}{4} x^{3}$.

Step 1: Write down the given information

In order to solve this problem, we need to identify and write down the given product and factor: Product: $$ \frac{1}{8} x^{4} y^{3} $$ Factor: $$ \frac{1}{2} x y^{3} $$

Step 2: Divide the product by the given factor

To find the other factor, we need to divide the product by the given factor. To do this, we can use the rule that \(\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}×\frac{d}{c}\). Similarly, we can use the rule for dividing exponents: \(\frac{a^m}{a^n}=a^{m-n}\). $$ \frac{(\frac{1}{8} x^{4} y^{3})}{(\frac{1}{2} x y^{3})} = (\frac{1}{8} x^{4} y^{3})×(\frac{2}{1} \frac{1}{x} \frac{1}{y^{3}}) $$

Step 3: Simplify the expression

Now, let's simplify the expression by multiplying the fractions and applying the rule for dividing exponents: $$ (\frac{1}{8} x^{4} y^{3})×(\frac{2}{1} \frac{1}{x} \frac{1}{y^{3}}) = \frac{1×2}{8}×x^{4-1} y^{3-3} $$

Step 4: Perform the operations and find the other factor

Perform the necessary operations to find the other factor: $$ \frac{1×2}{8}×x^{4-1} y^{3-3} = \frac{2}{8} x^{3} y^{0} $$ Since \(y^0 = 1\), we can simply write: $$ \frac{2}{8} x^{3} y^{0} = \frac{2}{8} x^{3} $$ Finally, simplified the fraction we get: $$ \frac{1}{4} x^{3} $$ So, the other factor is: $$ \frac{1}{4} x^{3} $$

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain an algebraic expression. In our exercise, \( \frac{1}{8} x^4 y^3 \) and \( \frac{1}{2} x y^3 \) are both algebraic fractions. Working with these fractions involves applying the same principles as numerical fractions with some additional rules related to variables.
To multiply algebraic fractions, simply multiply the numerators together and the denominators together. When dividing, invert the second fraction and multiply. In our exercise, we used this method by flipping \( \frac{1}{2} x y^3 \) to \( \frac{2}{1} \times \frac{1}{x} \times \frac{1}{y^3} \). This practice ensures that the problem converts from division to multiplication, making it easier to solve. Understanding algebraic fractions is crucial for solving polynomial expressions efficiently.

Exponent Rules

Working with exponents can seem tricky at first, but understanding a few fundamental rules makes it much more manageable. Exponents are used to express repeated multiplication and some basic rules apply.

• Product of Powers Rule: When multiplying two expressions with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents: \( a^m \div a^n = a^{m-n} \).
• Zero Exponent Rule: Any nonzero number raised to the power of zero is 1, \( a^0 = 1 \) where \( a e 0 \).

In the exercise, we applied these rules to simplify the polynomial terms. For instance, \( x^{4-1} \) simplifies to \( x^3 \), and \( y^{3-3} \) simplifies to \( y^0 = 1 \). Knowing how exponents work allows you to condense polynomial expressions with precision and is essential for multiplying and dividing polynomials.

Simplifying Expressions

Simplifying expressions is a crucial step in solving algebraic problems. It often requires applying both fraction operations and exponent rules.
The aim is to reduce the expression to its simplest form, which usually makes it easier to interpret or use in further calculations. In the given solution, starting with the expression \( \frac{1}{8} x^4 y^3 \times (\frac{2}{1} \times \frac{1}{x} \times \frac{1}{y^3}) \), we proceeded to simplify step by step:

• Multiply the fractions: \( \frac{2}{8} \) simplifies to \( \frac{1}{4} \).
• Use the quotient of powers rule for exponents: \( x^{4-1} = x^3 \) and \( y^{3-3} = y^0 = 1 \).
• Combine the results: \( \frac{1}{4} x^3 \).

Simplifying algebraic expressions helps you find the most streamlined solution. It's both a skill and an art, which becomes easier with practice. Mastering this process is vital for tackling more complex algebraic equations efficiently.