Problem 5 - Calculus Single Variable

Question

Simplify the given expression. $$ \left(\sqrt{11}^{\sqrt{2}}\right)^{\sqrt{2}} $$

Step-by-Step Solution

Answer

The simplified expression is 11.

Step 1: Understand the Power of a Power Property

Recall the property of exponents when raising a power to another power: \( (a^m)^n = a^{m \cdot n} \). This means you multiply the exponents when there is an exponent on both the base and the power.

Step 2: Apply the Power of a Power Property

Apply this power property to the expression \( \left(\sqrt{11}^{\sqrt{2}}\right)^{\sqrt{2}} \). The base is \( \sqrt{11} \) with an exponent of \( \sqrt{2} \) raised to another \( \sqrt{2} \). Thus, \( \left(\sqrt{11}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{11}^{\sqrt{2} \cdot \sqrt{2}} \).

Step 3: Simplify the Exponent

Simplify \( \sqrt{2} \cdot \sqrt{2} \). The product of a square root with itself is the number under the square root, so \( \sqrt{2} \cdot \sqrt{2} = 2 \).

Step 4: Write the Final Exponent

Substitute the simplified exponent back into the expression: \( \sqrt{11}^{2} \). Since the square root and square power cancel each other, \( \sqrt{11}^{2} = 11 \).

Key Concepts

Power of a Power

Understanding the concept of "power of a power" can be a game changer in simplifying complex expressions. Imagine you have a scenario where you not only have a base with an exponent, but this is also raised to another exponent. It might initially look intimidating—but there's a handy rule here. When you have something like

• defined by the form \[ (a^m)^n \]

— you just multiply the exponents:

• a multiplier shortcut that turns \[ a^{m \cdot n} \].

In practical terms, suppose you encounter

• a case such as \[ \left( \sqrt{11}^{\sqrt{2}} \right)^{\sqrt{2}} \].

You apply the power of a power rule here by multiplying the exponents. Thus, you have

• \[ \sqrt{11}^{\sqrt{2} \cdot \sqrt{2}} \]

— which simplifies further as you proceed through the problem.

Square Roots

Square roots provide a way to deload complicated exponents. The essence of a square root is finding a number which, when multiplied by itself, returns the original number.

• For example, the square root of 9 is 3 because \[ 3 \times 3 = 9 \].

But things get more interesting with expressions like

• \(\sqrt{11}\),

which are less straightforward since they can't be simplified into whole numbers. In the context of our expression, we find a square root raised to a square root—a somewhat unusual situation.Writing

• \(\sqrt{2} \times \sqrt{2} = 2\),

proves that the product of a square root with itself is simply the original number under the square root. This simplification is powerful when dealing with multiple square-root exponents.

Simplifying Expressions

Simplifying expressions is about stripping away layers of complexity to get to the heart of the expression. One main goal is to reduce the expression to its simplest form so that anyone can quickly understand the result.If you're facing nested exponents or roots, such as

• \(\left(\sqrt{11}^{\sqrt{2}}\right)^{\sqrt{2}}\),

you aim to simplify as much as possible by using fundamental properties like the power of a power. First, apply the multiplication of exponents:

• \(\sqrt{11}^{\sqrt{2} \cdot \sqrt{2}}\)

simplifies in this scenario to

• \(\sqrt{11}^{2}\).

Since squaring a square root effectively cancels out the square root, we're left with an elegant and simplified result,

• \(11\).