Problem 29 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 12^{\frac{5}{3}} \div 12^{\frac{2}{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to 12.

Step 1: Apply the Quotient of Powers Property

When dividing like bases (in this case, the base is 12), you can subtract the exponents. The property is given by \(a^m \div a^n = a^{m-n}\). Apply this to the expression \(12^{\frac{5}{3}} \div 12^{\frac{2}{3}}\). Thus, we have: \[ 12^{\frac{5}{3} - \frac{2}{3}} = 12^{\frac{3}{3}} \] This simplifies to \(12^1\).

Step 2: Simplify the Power

Now that you have simplified the power to \(12^1\), this simply equals 12, since any number to the power of 1 is the number itself. Hence, \(12^1 = 12\).

Key Concepts

Understanding Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers. For instance, the number 0.5 can be written as \( \frac{1}{2} \), and likewise, 3 can be rewritten as \( \frac{3}{1} \). These numbers encompass all fractions and entire sets of integers. Rational numbers are vital because:

• They include many common values like fractions and whole numbers.
• They can be positive, negative, or zero, providing a broad range of values.

In many math problems, like the one tackled here, complex expressions result in rational numbers after simplification. Understanding them helps in the expression of numerical results in a unified manner.

Simplifying Expressions

Simplifying mathematical expressions often involves reducing them to their most concise form using a set of rules or properties. This can make further calculations much easier and clearer. The process involves:

• Identifying like terms and combining them when dealing with polynomials or algebraic expressions.
• Using properties and operations, like canceling out common factors in fractions or using exponent rules to manage terms with exponents.

In the original problem, simplifying the expression through the division of exponential terms allowed us to reduce \( 12^{\frac{5}{3}} \div 12^{\frac{2}{3}} \) down to \( 12^1 \), or simply 12. Practicing simplification techniques improves problem-solving efficiency and clarity.

Applying the Quotient of Powers Property

The Quotient of Powers Property is a crucial rule when working with exponents. This property tells us how to manage expressions where similar bases are divided. The formula is \( a^m \div a^n = a^{m-n} \), indicating that you subtract the exponent in the denominator from the exponent in the numerator.This property simplifies expressions by reducing the operation to a single term:

• It simplifies calculations by reducing the amounts of multiplication involved.
• It reduces organization complexity, making the expression sleeker and easier to handle.

In our exercise, applying this property transformed \( 12^{\frac{5}{3}} \div 12^{\frac{2}{3}} \) into \( 12^{\frac{3}{3}} \), which equals 12. This property showcases the power of understanding and applying exponent rules effectively. Understanding the Quotient of Powers Property sharpens your ability to manipulate and solve expressions with exponents efficiently.