Problem 64 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified exponential expression is 1.

Step 1: Apply the zero-exponent rule

Applying the zero-exponent rule to \(\left(\frac{4 a^{-5} b^{3}}{12 a^{3} b^{-5}}\right)^{0}\), we can say that no matter what \(\frac{4 a^{-5} b^{3}}{12 a^{3} b^{-5}}\) equals to, as it is all being raised to the power of zero, it will simplify to one, because any nonzero number raised to the power of zero equals one.

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Step 5: Conclude with the answer

The simplified exponential expression is 1.

Key Concepts

Zero Exponent Rule

The Zero Exponent Rule is a simple, yet powerful concept in mathematics. Understanding it makes working with exponential expressions much easier. The rule states that any nonzero number raised to the power of zero is equal to one. This applies to complex expressions too. For example, when you see something like \( \left(\frac{4 a^{-5} b^{3}}{12 a^{3} b^{-5}}\right)^{0} \), don't let the fractions or negative exponents confuse you. Because the whole thing is inside a pair of parentheses and raised to the zero power, the answer is simply one. Here are a few points to remember:

• The base of the exponent doesn't matter (as long as it's not zero).
• Even complex expressions simplify to one when raised to the zero power.

Just think of it like a magic spell that simplifies everything to one, saving you a lot of calculation time!

Simplifying Expressions

Simplifying expressions can sometimes seem overwhelming, especially when variables and exponents are involved. However, breaking down the expression step by step can ease the process. Here's a simple approach:

• Identify and apply relevant rules, like the Zero Exponent Rule.
• Combine like terms and simplify fractions if applicable.

Consider the expression inside the parentheses from our exercise: \(\frac{4 a^{-5} b^{3}}{12 a^{3} b^{-5}}\). Before applying the Zero Exponent Rule, you would normally combine and simplify the terms:

• Divide the numerical coefficients: \( \frac{4}{12} = \frac{1}{3} \).
• Combine the powers of \(a\) by subtracting exponents: \( a^{-5 - 3} = a^{-8} \).
• Combine the powers of \(b\) in a similar way: \( b^{3 + 5} = b^{8} \).

Although these steps might feel unnecessary due to the zero exponent in this case, they're fantastic practice!

Exponential Expressions

Exponential expressions repeatedly occur in mathematics, characterizing growth processes or decay in various scientific fields. Here’s how you can understand exponential expressions better:

• Base and Exponent: The base is the number being multiplied, and the exponent tells how many times to multiply it.
• Negative Exponents: They represent the reciprocal of the base raised to the positive exponent. For example, \( a^{-5} = \frac{1}{a^{5}} \).
• Multiplying Exponents: When multiplying like bases, you add the exponents: \(a^{m} \times a^{n} = a^{m+n}\).
• Dividing Exponents: Conversely, divide like bases by subtracting the exponents: \(a^{m} \div a^{n} = a^{m-n}\).

Understanding these ideas forms the foundation for grasping more complex problems. In our exercise, we eventually used the properties of exponents to simplify terms, even before applying the Zero Exponent Rule. Mastery of exponential expressions makes mathematics feel like less of a mystery, and more logical.