Problem 79 - Elementary Algebra

Question

Assuming the bases are not zero, find the value of \(\frac{36 x^{10} y^{8} z^{3} w^{0}}{9 x^{5} y^{2} z}\).

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\frac{36 x^{10} y^{8} z^{3} w^{0}}{9 x^{5} y^{2} z}\) Answer: \(4x^5y^6z^2\)

Step 1: Divide the coefficients

First, we need to divide the coefficients 36 and 9, which results in 4: $$\frac{36}{9} = 4$$

Step 2: Apply the exponent rules for variables

Now, we'll apply the exponent rules to simplify the variables. Remember to subtract the exponents when dividing if the bases are the same: For \(x\): \(\frac{x^{10}}{x^{5}} = x^{10-5} = x^5\) For \(y\): \(\frac{y^{8}}{y^{2}} = y^{8-2} = y^6\) For \(z\): \(\frac{z^{3}}{z} = z^{3-1} = z^2\) For \(w\): \(w^{0} = 1\) Now, let's put everything together:

Step 3: Combine the simplified parts

Finally, we combine our simplified coefficients and variables: $$\frac{36 x^{10} y^{8} z^{3} w^{0}}{9 x^{5} y^{2} z} = 4x^5y^6z^2\times 1$$ So, the simplified expression is: $$4x^5y^6z^2$$

Key Concepts

Exponent Rules

Exponent rules are fundamental to simplifying expressions involving powers and roots. These rules are especially important when dealing with variables raised to a power, which is often the case in algebra. When dividing expressions with exponents, the key rule to remember is that you subtract the exponent of the denominator from the exponent of the numerator, provided the bases are the same.
Here are some essential exponent rules:

• Product of powers rule: When you multiply like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of powers rule: When you divide like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power rule: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Zero exponent rule: Any non-zero base raised to the zero power is 1: \(a^0 = 1\).

Using these rules can help simplify an expression significantly. In our example, we applied the quotient of powers rule to each of the variables, enabling straightforward simplification.

Polynomial Division

Polynomial division is a method used to divide polynomial expressions, similar to long division with numbers. When dividing polynomials, especially with variables, the exponent rules become crucial. It's important to approach the problem step-by-step, simplifying each term individually.
Here's a simple breakdown of the steps:

• Divide coefficients: First, divide the numbers in front (coefficients) as you would regular numbers. For instance, \(\frac{36}{9} = 4\).
• Divide like terms: For variables, apply the quotient of powers rule by subtracting exponents. Handle each variable separately to avoid mistakes.
• Combine results: Once all terms are simplified, put them together to finalize the expression.

In our specific problem, by following these steps, we divided the coefficients first, then proceeded with each variable using exponent rules. This led to a clean and simplified result.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to understand and work with. The process of simplification often involves a series of algebraic techniques, such as factoring, canceling common terms, and applying exponent rules.
Here’s a guide on simplifying expressions effectively:

• Look for common factors: If all terms share a common factor, factor it out before proceeding further.
• Use the right mathematical operations: Simplify terms using addition, subtraction, multiplication, and division as needed.
• Apply exponent rules: Use the rules discussed earlier to simplify terms involving exponents.
• Rewrite the expression: Once factors and operations are simplified, rewrite the expression neatly to make sure all terms are combined correctly.

Through these steps, the given algebraic expression was simplified efficiently by combining all individual simplifications. This approach makes handling complex expressions more manageable and less daunting.