Problem 82 - Algebra 2

Question

Exponents that are irrational numbers can be defined so that all the properties of rational exponents are also true for irrational exponents. Use those properties to simplify each expression. $$\frac{3^{3+\sqrt{5}}}{3^{1+\sqrt{5}}}$$

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Identify exponential properties

The first step is to recognize that the given expression falls under the property of exponential subtraction, \((a^{m}) / (a^{n}) = a^{(m-n)}\). This expresses that when you divide like bases, you subtract the exponents.

Step 2: Apply the property

Applying the aforementioned subtraction property to the given expression, it simplifies to: \(3^{(3+\sqrt{5})-(1+\sqrt{5})}\).

Step 3: Simplify the expression

The expression simplifies further when the like terms are subtracted. Subtract \(1+\sqrt{5}\) from \(3+\sqrt{5}\) leaves us with 2. Therefore, the expression turns into \(3^2\).

Step 4: Final simplification

The final step is to calculate the value of \(3^2\) which is 9. Hence, the simplified expression is 9.

Key Concepts

Exponential Properties

Understanding exponential properties is crucial for solving expressions involving exponents. The properties are like a set of rules that make handling exponents simpler and more predictable. One of these essential rules is the property of subtraction:

• When dividing terms with the same base, subtract the exponents: If you have an expression like \((a^m) / (a^n)\), you can simplify it to \(a^{(m-n)}\).

This concept helps make sense of expressions that might initially seem complex, particularly when working with both rational and irrational exponents. No matter what the exponents are, this rule stays consistent.
This property allows us to simplify expressions by reducing the number of exponents and focusing on just the base and the difference between the exponents.

Simplifying Expressions

Simplifying an expression is about breaking it down into a simpler form while maintaining the same value. It's like tidying up math to make it more manageable and easy to work with.In the specific exercise, simplifying the expression \(\frac{3^{3+\sqrt{5}}}{3^{1+\sqrt{5}}}\) involves a few key steps:

• Identify the base and exponents: Notice that both the numerator and denominator have the same base, which is 3.


• Apply the subtraction property: As discussed earlier, divide the two expressions by subtracting their exponents: \(3^{(3+\sqrt{5}) - (1+\sqrt{5})}\).


• Simplify the exponents: This means doing a simple arithmetic operation where you subtract \(1+\sqrt{5}\) from \(3+\sqrt{5}\) to get 2.

Finally, the expression simplifies to \(3^2\), which equals 9. The process of simplifying can often be repeated with different expressions, following similar steps to ensure clarity.

Rational Exponents

Rational exponents, unlike whole numbers, can include fractions or other numbers like irrational numbers. However, they follow the same exponents' rules as whole numbers, making calculations consistent and reliable. To better understand rational exponents:

• Definition: A rational exponent represents both roots and powers of numbers. For instance, \(a^{1/2}\) represents the square root of \(a\), and \(a^{3/2}\) implies the square root of \(a^3\).


• Calculation Confidence: Even when dealing with complicated expressions, knowing they follow the same rules gives confidence in handling them as comfortably as whole-number exponents.

For exponential expressions like the given exercise, checking the rational accuracy of your calculations becomes simpler when you know these underlying rules. Understanding rational exponents allows students to handle complex mathematical scenarios with ease, turning what seems tricky into manageable math. Working with rational exponents is essential in advanced mathematics as they often connect to real-world scenarios needing precise and clear solutions.