Solved Problems: Power of a Product and Power of a Quotient for High School

📚 Power of a Product: Understanding the Basics

The power of a product rule states that when you raise a product to a power, you distribute the exponent to each factor within the product. Mathematically, this is represented as:

$(ab)^n = a^n b^n$

This means that if you have a product, say $ab$, raised to the power of $n$, it is the same as $a$ raised to the power of $n$ times $b$ raised to the power of $n$.

📜 Historical Context

The understanding and formalization of exponent rules, including the power of a product, evolved over centuries. Early mathematicians in ancient civilizations like Babylon and Greece developed initial concepts of exponents. However, the symbolic notation and generalized rules we use today were refined during the development of algebra in the medieval Islamic world and later in Europe during the Renaissance.

🔑 Key Principles

• 🔢 Distribute the Exponent: The most crucial step is to correctly distribute the exponent to each factor inside the parentheses.
• ➕ Apply to Each Factor: Ensure that every factor within the product is raised to the given power.
• 💡 Simplify: After distributing the exponent, simplify each term individually.

📝 Real-world Examples

Example 1: Simple Application

Simplify $(2x)^3$

Solution:

$(2x)^3 = 2^3 * x^3 = 8x^3$

Example 2: Multiple Variables

Simplify $(3ab^2)^2$

Solution:

$(3ab^2)^2 = 3^2 * a^2 * (b^2)^2 = 9a^2b^4$

Example 3: Numerical Coefficients and Variables

Simplify $(4x^2y)^3$

Solution:

$(4x^2y)^3 = 4^3 * (x^2)^3 * y^3 = 64x^6y^3$

➗ Power of a Quotient: Understanding the Basics

The power of a quotient rule states that when you raise a quotient to a power, you distribute the exponent to both the numerator and the denominator. Mathematically, this is represented as:

$(\frac{a}{b})^n = \frac{a^n}{b^n}$

This means that if you have a quotient, say $\frac{a}{b}$, raised to the power of $n$, it is the same as $a$ raised to the power of $n$ divided by $b$ raised to the power of $n$, provided that $b$ is not zero.

📜 Historical Context

Similar to the power of a product, the power of a quotient rule has roots in the historical development of algebraic notation and exponent manipulation. The formalization of these rules allowed mathematicians to simplify and solve more complex equations involving fractions and exponents.

🔑 Key Principles

• ➗ Distribute the Exponent: Distribute the exponent to both the numerator and the denominator.
• 📊 Apply to Both: Ensure that both the numerator and the denominator are raised to the given power.
• ✨ Simplify: Simplify both the numerator and the denominator separately after distributing the exponent.

📝 Real-world Examples

Example 1: Simple Application

Simplify $(\frac{x}{2})^4$

Solution:

$(\frac{x}{2})^4 = \frac{x^4}{2^4} = \frac{x^4}{16}$

Example 2: Variables and Coefficients

Simplify $(\frac{2x}{y})^3$

Solution:

$(\frac{2x}{y})^3 = \frac{(2x)^3}{y^3} = \frac{2^3 * x^3}{y^3} = \frac{8x^3}{y^3}$

Example 3: Complex Fractions

Simplify $(\frac{3a^2}{b})^2$

Solution:

$(\frac{3a^2}{b})^2 = \frac{(3a^2)^2}{b^2} = \frac{3^2 * (a^2)^2}{b^2} = \frac{9a^4}{b^2}$

📊 Practice Quiz

Simplify the following expressions:

1. $(5x^3)^2$
2. $(\frac{4}{y^2})^3$
3. $(2ab^3)^4$
4. $(\frac{3x^2}{2y})^2$
5. $(7m)^2$

✅ Answers

1. $25x^6$
2. $\frac{64}{y^6}$
3. $16a^4b^{12}$
4. $\frac{9x^4}{4y^2}$
5. $49m^2$