Problem 11 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$-\left(\frac{p^{4}}{q}\right)^{2}$$

Step-by-Step Solution

Answer

\(-\frac{p^8}{q^2}\)

Step 1: Apply the Power Rule

The given expression is \(-\left(\frac{p^4}{q}\right)^2\). First, apply the power of a quotient rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This gives us \(-\frac{(p^4)^2}{q^2}\).

Step 2: Simplify the Numerator

Simplify \((p^4)^2\) using the power of a power rule: \((a^m)^n = a^{m \cdot n}\). This becomes \(p^{4 \cdot 2} = p^8\).

Step 3: Write the Final Expression

Combine the simplified numerator and the denominator: the simplified expression is \(-\frac{p^8}{q^2}\).

Key Concepts

Power Rule

The Power Rule is a key tool in simplifying expressions involving exponents. When you see an exponent raised to another exponent, you apply this rule. It states that \((a^m)^n = a^{m imes n}\). This means you multiply the exponents together.

For instance, if you have \((p^4)^2\), you multiply \(4\) by \(2\) to get \(p^{8}\).

It's very useful in making complex expressions more manageable by reducing the number of operations needed.

Power of a Quotient Rule

The Power of a Quotient Rule helps simplify expressions where a fraction is raised to an exponent. According to this rule, when you have a fraction with an exponent like \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), you apply the exponent to both the numerator and the denominator separately.

So, for a fraction like \(\left(\frac{p^4}{q}\right)^2\), you handle it by calculating \(\frac{(p^4)^2}{q^2}\).

This is very effective for breaking down and understanding expressions involving fractions and exponents.

• Great for dealing with fractions raised to a power.
• Makes division operations easier.

Exponents

Exponents represent repeated multiplication. They are crucial in algebra for making lengthy multiplication easier to work with. For example, \(2^3\) means \(2 \times 2 \times 2\), which equals \(8\).

Exponents are used extensively in simplifying expressions and solving equations. They follow specific rules, such as the Power Rule and the Power of a Quotient Rule.

• They show how many times to multiply a base by itself.
• They simplify writing and calculating large numbers.

When working with exponents, always remember to follow the order of operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. These can include addition, subtraction, multiplication, and division, alongside exponents.

Expressions like \(-\frac{p^8}{q^2}\) show how algebra helps represent complex ideas simply and effectively.

In algebra, simplifying expressions by applying rules such as the Power Rule is fundamental. This helps in solving equations and understanding relationships between variables. Working through problems step-by-step makes algebra manageable and less intimidating.

• Essential for problem-solving.
• Represents real-world situations with variables.
• Makes abstract concepts concrete.

When simplifying, ensure you apply each rule correctly to reach a valid solution.