Problem 29 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ 8^{-4 \sqrt{3}} $$

Step-by-Step Solution

Answer

The expression \(8^{-4\sqrt{3}}\) can be rewritten as \(\frac{1}{8^{4\sqrt{3}}}\).

Step 1: Understand the Expression with Negative Exponent

When a base is raised to a negative exponent, it can be rewritten as the reciprocal of the base raised to the opposite positive exponent. Thus, an expression like \(8^{-4\sqrt{3}}\) can be rewritten using positive exponents.

Step 2: Rewrite with Positive Exponents

To rewrite \(8^{-4\sqrt{3}}\) using positive exponents, apply the rule of negative exponents: \(a^{-n} = \frac{1}{a^n}\). So, \(8^{-4\sqrt{3}}\) becomes \(\frac{1}{8^{4\sqrt{3}}}\).

Step 3: Check for Simplification

Since there are no like terms or simplifications within \(8^{4\sqrt{3}}\), \(\frac{1}{8^{4\sqrt{3}}}\) is the simplified form of the expression.

Key Concepts

Positive Exponents

Positive exponents are straightforward and intuitive. When we say a number has a positive exponent, it simply means multiplying the base number by itself as many times as indicated by the exponent. For instance, the expression \(8^3\) means \(8 \times 8 \times 8\), which results in 512.

• The exponent tells us **how many times to multiply the base together.**
• A positive exponent results in larger numbers if the base is more than 1.
• An exponent of 1 leaves the base unchanged, while an exponent of 0 transforms it into 1.

In any expression, rewriting all parts using positive exponents usually makes the expression simpler to understand and manipulate mathematically. Transitioning from negative to positive exponents can unravel confusion, especially in complex evaluations.

Reciprocal

The reciprocal of a number is like flipping it upside down. More formally, for any non-zero number \(a\), its reciprocal is \(\frac{1}{a}\). This is a foundational concept in math, especially when dealing with fractions and exponents.
Think of negative exponents as a cue to find the reciprocal. When you see a negative exponent, like in \(8^{-4\sqrt{3}}\), it means you need to take the reciprocal of the base, \(8\), and change the exponent's sign to positive.

• A negative exponent means "**flip the base to its reciprocal"** before applying the exponent.
• For instance, \(x^{-n}\) means \(\frac{1}{x^n}\).
• This operation helps convert an expression into a form with positive exponents.

Understanding reciprocals gives clarity when dealing with negative exponents, paving the way for clearer and more efficient solutions in algebraic problems.

Simplification

Simplification is the process of rewriting an expression in its simplest or most understandable form. It often involves reducing fractions, combining like terms, or changing exponents.
When dealing with an expression like \(\frac{1}{8^{4\sqrt{3}}}\), simplification checks include:

• Ensuring there are no smaller like terms to combine.
• Verifying if the base or exponent can be further simplified.
• Checking if there are patterns or operations to condense the expression.

In many cases, mathematical expressions involve numbers or terms that can't be simplified beyond what's visible, as is the case with \(8^{4\sqrt{3}}\). At this point, it's crucial to recognize the expression is as simple as it gets, which helps avoid unnecessary computations and clarifies the expression's form.