Problem 68 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{a^{0} b^{0}}{c^{0}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the product rule and quotient rule of exponents: $\frac{a^{0} b^{0}}{c^{0}}$. Answer: 1

Step 1: Use Power of 0 Rule

: Recall that any nonzero number raised to the power of 0 is equal to 1. Based on this property, we can simplify the given expression as follows: $$ \frac{a^{0} b^{0}}{c^{0}} = \frac{1*1}{1} $$

Step 2: Simplify

: Now, we can simplify the expression: $$ \frac{1*1}{1} = \frac{1}{1} $$

Step 3: Final Answer

: Since the fraction is equal to 1, the simplified expression is: $$ \frac{a^{0} b^{0}}{c^{0}} = 1 $$

Key Concepts

Product Rule

The product rule for exponents is an essential tool when working with expressions involving multiplication of the same base. Simply put, it states that when you multiply two exponents with the same base, you can add their exponents together. For instance, if you have \( a^m \times a^n \), this can be simplified to \( a^{m+n} \). Here’s how you can think about it:

• The base \(a\) stays the same because you are multiplying the same number multiple times.
• Simply add the exponents \(m\) and \(n\) together to find the new exponent.

Remember, the product rule only works with identical bases. When you encounter different bases, the rule cannot be applied, and each base will need to be handled separately. Practice makes perfect, so try using this rule on a variety of expressions to become confident with the process.

Quotient Rule

The quotient rule is very similar to the product rule, except it applies to division instead of multiplication. This rule tells us that when dividing exponents with the same base, you should subtract the exponent of the denominator from the exponent of the numerator. For example, given the expression \( \frac{a^m}{a^n} \), it simplifies to \( a^{m-n} \).

• Keep the base \(a\) unchanged.
• Subtract the exponent in the denominator \(n\) from the exponent in the numerator \(m\).

This rule is particularly useful when you need to simplify complex fractions that involve exponents. Just like with the product rule, the bases must match for the quotient rule to apply. It’s also important to be careful with subtraction as it is order-sensitive and might result in negative exponents, which have their own set of rules.

Power of Zero

The power of zero rule is a very simple yet crucial concept in the study of exponents. It states that any non-zero number raised to the power of zero is always equal to one. Mathematically, this looks like \( a^0 = 1 \), provided \( a eq 0 \).

• This is a universal rule and can be applied to any non-zero number or variable.
• It simplifies expressions by turning them into multiplication by one, which leaves the other factors unchanged.

In practical use, the power of zero helps to reduce complicated expressions by simplifying parts of them. For example, in the expression \( \frac{a^0 b^0}{c^0} \), each part of the fraction becomes one, turning the entire expression into simply one. This rule is one of the pillars of working with exponents, making even the most daunting expressions much more manageable.