Problem 4 - Calculus Single Variable

Question

Simplify the given expression. $$ \left(8^{\sqrt{3}} \cdot 4^{\sqrt{7}}\right) / 2^{\pi} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2^{3\sqrt{3} + 2\sqrt{7} - 2\pi} \).

Step 1: Break down the expression into its components

The expression given is \( \left(8^{\sqrt{3}} \cdot 4^{\sqrt{7}}\right) / 2^{\pi} \). This can be rewritten as \( \frac{8^{\sqrt{3}}}{2^{\pi}} \cdot \frac{4^{\sqrt{7}}}{2^{\pi}} \) by separating the terms.

Step 2: Express bases with a common base

Note that 8 and 4 can be expressed as powers of 2: \( 8 = 2^3 \) and \( 4 = 2^2 \). Thus, \( 8^{\sqrt{3}} = (2^3)^{\sqrt{3}} = 2^{3\sqrt{3}} \) and \( 4^{\sqrt{7}} = (2^2)^{\sqrt{7}} = 2^{2\sqrt{7}} \).

Step 3: Use properties of exponents to simplify the expression

Now, the expression becomes \( \frac{2^{3\sqrt{3}}}{2^{\pi}} \cdot \frac{2^{2\sqrt{7}}}{2^{\pi}} \). Using the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \), we have \( 2^{3\sqrt{3} - \pi} \cdot 2^{2\sqrt{7} - \pi} \).

Step 4: Combine exponents with the same base

Combine the exponents having the same base: \( 2^{3\sqrt{3} - \pi + 2\sqrt{7} - \pi} \). Simplifying the exponent gives \( 2^{3\sqrt{3} + 2\sqrt{7} - 2\pi} \).

Key Concepts

Exponential expression simplification

When we encounter an exponential expression like \( \left(8^{\sqrt{3}} \cdot 4^{\sqrt{7}}\right) / 2^{\pi} \), simplifying it involves several steps. The goal is to make the expression easier to understand or solve by reducing it to a simpler form.

Start by breaking the expression into its components. For this exercise, it means rewriting it to separate each base and its corresponding exponent. You can express it as \( \frac{8^{\sqrt{3}}}{2^{\pi}} \cdot \frac{4^{\sqrt{7}}}{2^{\pi}} \).

This separation helps us focus on each part independently and sets the stage for further simplification steps. By tackling the expression this way, each section can be converted or simplified based on specific rules and properties of exponents.

Properties of exponents

The properties of exponents are essential tools when simplifying expressions. They allow us to manipulate and reduce complex expressions into more manageable forms. Here are some key properties:

• Power of a power rule: \((a^m)^n = a^{m \cdot n}\)
• Quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Product of powers rule: \(a^m \cdot a^n = a^{m+n}\)

With these properties, you can take a complicated expression like \( \frac{2^{3\sqrt{3}}}{2^{\pi}} \cdot \frac{2^{2\sqrt{7}}}{2^{\pi}} \) and use the quotient rule to simplify each fraction by subtracting exponents. This procedure provides a cleaner and more straightforward expression, making it easier to further work with or understand the mathematical relationship.

Common bases

Identifying common bases in an expression is a helpful step towards simplification. A base refers to the number that is being repeatedly multiplied in an exponent. For instance, in \( 2^3 \) and \( 2^5 \), 2 is the common base.

When the bases in an expression are the same, it opens up opportunities to combine the exponents using properties of exponents. In our exercise, the numbers 8 and 4 can both be expressed with a common base of 2:

• 8 as \(2^3\)
• 4 as \(2^2\)

Therefore, transforming \(8^{\sqrt{3}}\) into \(2^{3\sqrt{3}}\) and \(4^{\sqrt{7}}\) into \(2^{2\sqrt{7}}\), allows us to rewrite the original expression as a single base of 2. Now, you can consolidate the exponents through addition or subtraction, following the rules of exponents, and create a simpler form. This is a powerful technique in exponential expression simplification.