Problem 86 - 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. $$\left(3^{2+\sqrt{2}}\right)^{2-\sqrt{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 9

Step 1: Application of the exponentiation property

We should apply the property of exponentiation in the form of \((a^m)^n = a^{m \cdot n}\) to greatly simplify the problem. Hence, the expression \(\left(3^{2+\sqrt{2}}\right)^{2-\sqrt{2}}\) becomes \(3^{(2+\sqrt{2})(2-\sqrt{2})}\)

Step 2: Distribute and Simplify

Distribute and simplify this part: \((2+\sqrt{2})(2-\sqrt{2})\). This is a straightforward application of the difference of squares, yielding \(2^{2}-(\sqrt{2})^{2}\), which simplifies to \(4-2=2\)

Step 3: Final Simplification

Substitute the result of Step 2 back into our expression gives us \(3^2\), which equals to \(3*3 = 9\)

Key Concepts

Properties of Exponents

Exponent properties are powerful tools that simplify expressions by manipulating the powers of numbers or variables. These properties help maintain consistency when working with exponential expressions. Key properties include:

• Product of Powers: When you multiply two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When you raise a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Power of a Product: Distribute the power across the products as \((ab)^m = a^m b^m\).
• Quotient of Powers: When dividing with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

These properties are valid even when the exponent is an irrational number. This is crucial for solving problems like the given exercise where the exponent contains irrational numbers.

Simplifying Expressions

Simplifying expressions with exponents involves applying the properties of exponents to convert the expression into its simplest form. The goal is to reduce complexity while ensuring that the expression remains equivalent to the original one.

When given an expression like \(\left(3^{2+\sqrt{2}}\right)^{2-\sqrt{2}}\), the first step is often to use the **Power of a Power** property to simplify the expression. In this case, it means transforming the expression into \(3^{(2+\sqrt{2})(2-\sqrt{2})}\).

The process involves carefully handling irrational numbers within the exponent and following arithmetic simplification methods to achieve a "clean" result. Simplification ensures the expression is easier to understand and solve, often transforming it into a more familiar number.

Difference of Squares

The difference of squares is a useful algebraic identity that simplifies products of binomials. It states that \((a+b)(a-b) = a^2 - b^2\). This identity can break down complex expressions into simpler terms.

In our exercise, we encounter \((2+\sqrt{2})(2-\sqrt{2})\). Using the difference of squares, this simplifies directly into \(2^2 - (\sqrt{2})^2\), which further resolves to \(4-2\) and results in \(2\).

Understanding this identity allows you to efficiently simplify complicated expressions without unnecessary steps, making problems like this quicker to solve.

Exponentiation Property

Exponentiation often involves breaking down complex expressions using established properties. In our exercise, the **Power of a Power** property plays a central role by transforming complex exponents into simpler terms through multiplication.

Applying this to \(\left(3^{2+\sqrt{2}}\right)^{2-\sqrt{2}}\), we perform multiplication of exponents to get \(3^{(2+\sqrt{2})(2-\sqrt{2})}\). This reveals the simplicity hidden within multi-layered exponent expressions.

Exponentiation properties help ensure expressions, even with irrational numbers, are handled seamlessly. The final result, \(3^2\), illustrates how such properties simplify the process and make it easier to achieve a clear, manageable result like \(9\).