Problem 81 - 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. $$\left(7^{\sqrt{2}}\right)^{\sqrt{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(7^2\), which equals 49.

Step 1: Identify the base and exponent

To simplify the expression \(\left(7^{\sqrt{2}}\right)^{\sqrt{2}}\), first identify the base, which is 7, and the exponents, which are \(\sqrt{2}\). The exponent \(\sqrt{2}\) occurs twice, so they are the \(m\) and \(n\) in our rule \((a^m)^n = a^{mn}\).

Step 2: Use the properties of exponents to simplify the expression

Next, use the property of exponents that \((a^m)^n = a^{mn}\) to simplify the expression. Substitute 7 for \(a\), \(\sqrt{2}\) for \(m\), and \(\sqrt{2}\) for \(n\) to rewrite the expression as \(7^{(\sqrt{2}*\sqrt{2})}\).

Step 3: Simplify the exponent

The product of \(\sqrt{2}\) and \(\sqrt{2}\) is 2. Therefore, we can simplify the expression further to \(7^2\).

Key Concepts

Understanding Properties of Exponents

Exponents are powerful mathematical tools that help us simplify and manipulate expressions. One of the key properties of exponents is the Power of a Power property. This property says that when you have an exponent raised to another exponent, like \((a^m)^n\), you can simplify it to \(a^{mn}\). This means you multiply the exponents together. This property is simple but extremely helpful in understanding more complex expressions.

• Let’s say you have \( (3^4)^2 \). According to the property, you multiply 4 and 2 to get \(3^8\).
• This approach works even if your exponents are irrational numbers, like \(\sqrt{2}\).

Understanding these properties ensures that you can work easily with both rational and irrational exponents. It keeps complex operations manageable by consistently applying simple rules.

Decoding Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal parts. A classic example is \(\pi\) or \(\sqrt{2}\). While irrational, these numbers often appear in expressions with exponents.
Even if the number is complex, when it comes into play as an exponent, such as \(7^{\sqrt{2}}\), you treat it just like any other exponent in relation to the properties of exponents.

• In the exercise, the number \(\sqrt{2}\) was used as the exponent.
• By understanding the properties, irrational exponents can be managed in the same way as rational ones.

So, even when faced with irrational numbers, we can use exponential rules to simplify and understand expressions effectively.

Simplifying Expressions

Simplifying expressions means making them easier to read, understand, and evaluate. There are many strategies to remove complexities from expressions, but with exponents, it's all about applying the right properties.
For instance, in \(\left(7^{\sqrt{2}}\right)^{\sqrt{2}}\), applying the Power of a Power property allowed simplification into \(7^{\sqrt{2}\times\sqrt{2}}\). By performing the multiplication, which results in 2, you further simplify it to \(7^2\).

• Each step makes the expression easier to manage.
• Ultimately, you end up with a straightforward result: \(49\).

By following these steps and using foundational properties, complex expressions involving irrational numbers become more understandable and easier to work with.