Problem 85 - 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. $$\frac{1}{9^{\frac{1}{\sqrt{2}}}}$$

Step-by-Step Solution

Answer

The simplified expression for \( \frac{1}{9^{\frac{1}{\sqrt{2}}}} \) is \(3^{-\sqrt{2}}\)

Step 1: Rewrite the Expression

The given expression is \( \frac{1}{9^{\frac{1}{\sqrt{2}}}} \). We can rewrite this using the rule that \( \frac{1}{a^n} = a^{-n} \), to obtain \( 9^{-\frac{1}{\sqrt{2}}} \)

Step 2: Rewrite the Base

Next, Rewrite the base 9 as \(3^2\), to get \( (3^2)^{-\frac{1}{\sqrt{2}}} \)

Step 3: Apply Exponential Rule

Now apply the rule \( (a^{m})^n = a^{mn} \). This gives \( 3^{2 \cdot -\frac{1}{\sqrt{2}}} = 3^{-\sqrt{2}}\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with complex expressions, especially those involving irrational exponents. Here are some key properties:

• Product of Powers Property: If you multiply two exponents with the same base, you add the exponents. Mathematically, this is written as: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Property: When raising an exponent to another power, multiply the exponents: \( (a^m)^n = a^{mn} \).
• Power of a Product Property: When taking an exponent of a product, distribute the exponent: \( (ab)^n = a^n b^n \).
• Negative Exponent Property: A negative exponent represents a reciprocal: \( a^{-n} = \frac{1}{a^n} \).

These properties help simplify expressions by transforming them into easier forms for calculation. Even when exponents are irrational, these rules apply, assisting in the breakdown and simplification of expressions.

Simplifying Expressions

Simplifying expressions, particularly those involving exponents, requires a systematic approach. It involves rewriting expressions in their simplest form. This is often done by

• Converting expressions using properties of exponents like negative exponents or power of powers.
• Rewriting bases if possible, such as converting 9 into \(3^2\).
• Combining like terms or simplifying fractional exponents to clearer base-exponent forms.

This process makes complex expressions more manageable. For instance, in simplifying \( \frac{1}{9^{\frac{1}{\sqrt{2}}}} \), we apply the negative exponent property, turning the fraction into a more straightforward base with a simpler power expression \( 9^{-\frac{1}{\sqrt{2}}} \). Then, further decomposing using known exponent properties leads us to \(3^{-\sqrt{2}}\).

Irrational Numbers

Irrational numbers are numbers that can't be expressed as simple fractions. They are numbers with non-repeating, non-terminating decimal points, such as \( \sqrt{2} \) or \( \pi \). These numbers are crucial when delving into exponents as they open a new realm of calculations and expressions.

• An example of such a number when working with exponents is \( \sqrt{2} \), where the decimal form is endlessly non-repetitive.
• Using irrational numbers within exponents results in forms like \(a^{\sqrt{2}}\), which need to be evaluated carefully, often using approximation methods or leaving them in an exact analytical form for exact solutions.

Understanding irrational numbers helps in grasping how they function within the laws of exponents, leading us to more precise and refined expressions and solutions.

Exponential Rules

Exponential rules form the foundation for understanding expressions involving powers. They include a set of operational guidelines that make it easier to handle any exponential expressions.

• Base Rewriting: Often, numbers can be rewritten in terms of smaller bases, like changing 9 to \(3^2\). This allows us to simplify expressions steadily.
• Exponent Operations: These include adding and subtracting exponents, and converting negative to positive exponents by utilizing reciprocals.
• Rationalizing to Solve: Irrational exponents require careful handling; exponential rules dictate precise ways to rationalize or express such exponents in most manageable forms.

In our exercise, knowing these rules allowed the step-by-step conversion from \( 9^{\frac{1}{\sqrt{2}}} \) to \( 3^{-\sqrt{2}} \), each change consistent with core exponential rules.