Problem 106 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{\left(2 x^{6} y^{2}\right)^{5}}{-32 x^{20} y^{10}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-x^{10}\).

Step 1: Apply Power to Each Factor

The expression to simplify is \( \frac{(2x^6y^2)^5}{-32x^{20}y^{10}} \). Start by applying the power to each term inside the parentheses in the numerator: \((2x^6y^2)^5 = 2^5(x^6)^5(y^2)^5\). This results in:\[2^5x^{30}y^{10}\].

Step 2: Evaluate Powers

Calculate the powers: \(2^5 = 32\), so the expression in the numerator becomes \(32x^{30}y^{10}\).

Step 3: Simplify Coefficients

Now, divide the coefficients in the expression \(\frac{32}{-32}\) which gives \(-1\).

Step 4: Simplify Exponents Using Quotient Rule

Apply the quotient rule of exponents \(\frac{x^m}{x^n} = x^{m-n}\) to the terms: \(\frac{x^{30}}{x^{20}} = x^{10}\) and \(\frac{y^{10}}{y^{10}} = y^{0} = 1\). So the expression becomes \(-1 \cdot x^{10} \cdot 1 = -x^{10}\).

Step 5: Write the Final Answer

The simplified expression is \(-x^{10}\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is like tidying up a room. It involves breaking down expressions to their simplest form. The goal is to combine like terms and use mathematical rules to make expressions easier to work with.
To simplify an expression, you often need to:

• Remove parentheses by applying exponent rules or distributing terms.
• Combine like terms, which are terms that have the same variables raised to the same power.
• Apply arithmetic operations to constants and coefficients.

Whenever you simplify, each step should make the expression easier to compute or more comprehensible. And that's exactly what we've done in solving this problem.

Exponent Rules

Exponent rules are fundamental to simplifying expressions, particularly when dealing with powers. They provide a structured way to navigate expressions involving numbers raised to powers.
Here are a few key rules:

• Product of Powers Rule: If you multiply two powers with the same base, you add their exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• Zero Exponent Rule: Any non-zero number raised to the power of zero is one: \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent indicates that the base should be reciprocated: \( a^{-m} = \frac{1}{a^m} \).

These rules provide consistency and precision, making algebraic manipulation simpler and more efficient.

Power of a Product

Understanding the "power of a product" rule is crucial when simplifying expressions where a power applies to a product inside parentheses. This rule states that when you have a product inside parentheses raised to a power, you apply the power to each factor individually.
This can be expressed mathematically as:\[(a \times b)^{n} = a^{n} \times b^{n}\]Using this rule helps simplify complex expressions, ensuring each part of the product is correctly powered, leading to simpler, cleaner expressions.
In our original problem, we applied this rule to \((2x^6y^2)^5\), distributing the exponent 5 to each factor: \(2^5(x^6)^5(y^2)^5\). By dealing with each component separately, the simplification becomes much more straightforward.

Quotient Rule

The quotient rule is another essential tool used while simplifying expressions, especially when dividing like bases with different exponents.
The rule states:\[\frac{a^{m}}{a^{n}} = a^{m-n}\]This says when you divide two expressions with the same base, you subtract the exponents.
The quotient rule simplifies expressions significantly by reducing terms and aligning similar bases.

• In our exercise, we used this rule to simplify expressions like \( \frac{x^{30}}{x^{20}} = x^{10} \), where subtracting the exponents 30 and 20 showed the reduced power of 10.
• Similarly, \( \frac{y^{10}}{y^{10}} = y^{0} = 1 \), exemplifies the power of simplification where terms cancel each other out completely.

With practice, recognizing when to apply the quotient rule during calculations becomes second nature, and it will make handling expressions easier.