Problem 63 - College Algebra

Question

Simplify each exponential expression $$ \left(\frac{3 a^{-5} b^{2}}{12 a^{3} b^{-4}}\right)^{0} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is 1.

Step 1: Identify the Exponential Rule

The first step is to identify the applicable exponential rule. In this case, anything (except zero) raised to the power of 0 equals 1. Therefore, the base, which is the complex fraction doesn't even matter in this particular case because of the power of zero.

Step 2: Apply the Exponential Rule

Next, apply the exponential rule to the expression. Since \((\frac{3 a^{-5} b^{2}}{12 a^{3} b^{-4}})\) is raised to the power of 0, it simplifies to 1 based on the previously mentioned rule of exponents.

Key Concepts

Exponential Rule

An exponential rule is a mathematical principle involving the operations on expressions with exponents or powers. In this context, an exponent tells you how many times a number, which is the base, is multiplied by itself.
The basic exponential rules help simplify complex expressions. The key rule here involves expressions where the exponent is 0.
Remember, if you have an expression like \((x^n)\), the rules change based on the value of \(n\).

• For positive \(n\), \(x^n\) becomes \(x \times x \times x \ldots\) (\(n\) times).
• For \(n = 0\), \(x^0\) always equals 1, provided \(x eq 0\).
• The power of zero rule is particularly powerful when simplifying expressions because it allows us to reduce complex terms in a straightforward manner.

Adopting this understanding will help you tackle any exponential expression effectively.

Power of Zero

The power of zero is one of the most misunderstood yet incredibly simple rules in mathematics. If you encounter any non-zero number or expression raised to the power of zero, the answer is always 1. This concept simplifies calculations greatly.
Why is this the case? Imagine you have any number \(x\) raised to various powers.

• \(x^1 = x\)
• \(x^2 = x \times x\)
• \(x^3 = x \times x \times x\)

Now, when you continue this logic:
\(x^3\) divided by \(x^3\) is 1 by definition. This is still true when you subtract exponents: \(x^{3-3} = x^0 = 1\). Therefore, any expression or number except zero raised to the power of zero equals 1.
This rule is critical, especially for simplifying expressions like the one in our exercise: \(\left(\frac{3a^{-5}b^{2}}{12a^{3}b^{-4}}\right)^0 = 1\). This power of zero rule renders all internal complexity of the expression unnecessary in the final result.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This makes complex mathematical equations easier to handle and solve.
Consider the expression \(\left(\frac{3a^{-5}b^{2}}{12a^{3}b^{-4}}\right)\). Normally, you would handle each part separately, simplifying fraction coefficients, and applying exponent rules.

• Coefficients \(3\) and \(12\) would usually simplify to \(\frac{1}{4}\).
• Exponents: Subtract exponents with same base (according to rules like \(a^m \div a^n = a^{m-n}\)).

However, because the entire expression is raised to the 0 power, it defaults to 1 immediately, removing the need for further simplification steps.
Understanding how to simplify expressions using the power of zero promotes efficiency in solving mathematical problems. It allows focus on the essential calculation rules while discarding non-crucial complexities in a fraction of the time.