Problem 50 - 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. $$ \frac{6^{4}}{6^{3}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is 6.

Step 1: Identify the base and exponents in the given expression

In the given expression \(\frac{6^{4}}{6^{3}}\), the base is 6, the exponent in the numerator is 4, and the exponent in the denominator is 3.

Step 2: Apply the quotient rule of exponents

According to the quotient rule of exponents, we can rewrite the given expression as: $$ \frac{6^{4}}{6^{3}} = 6^{4-3} $$

Step 3: Evaluate the way of the expression

Now subtract the exponents: $$ 6^{4-3} = 6^{1} $$

Step 4: Write the final result

Since any number raised to the power of 1 is the same number, we have: $$ 6^{1} = 6 $$ So the simplified expression is 6.

Key Concepts

Understanding the Product Rule of Exponents

When dealing with exponents, the product rule is a powerful tool. It helps simplify expressions where multiplication of like bases is involved. This rule states that when you multiply two numbers with the same base, you keep the base and add the exponents together.
For example, if you have the expression \(a^m \times a^n\), applying the product rule gives you \(a^{m+n}\).
This is because multiplying two powers with the same base means multiplying the numbers itself, which allows you to sum their exponents.

• The base remains the same.
• Add the exponents together.

Using this rule can significantly simplify the process of working through multiplication problems involving powers. Remember, it only applies to expressions with the same base.

Mastering the Quotient Rule of Exponents

The quotient rule for exponents is essential when simplifying expressions involving division of like bases. According to this rule, when you divide two numbers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.
Let's take an example with an expression \(\frac{a^m}{a^n}\). By applying the quotient rule, this expression simplifies to \(a^{m-n}\).
The steps are simple:

• Identify the base and the exponents.
• Subtract the exponent of the denominator from the exponent of the numerator.
• Keep the base unchanged.

By following these steps, you can swiftly simplify any expressions that involve dividing powers with the same base, just like the expression \(\frac{6^4}{6^3} = 6^{4-3} = 6^1 = 6\), which simplifies to 6.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves a combination of rules and steps that make the expressions easier to work with. Both the product and quotient rules are part of this process, allowing you to transform complex expressions into simpler ones.
When simplifying, the key is to:

• Identify and separate similar bases.
• Apply the appropriate exponent rules.
• Simplify any numerical coefficients if present.
• Combine like terms where necessary.

By changing the format of an expression, you reduce complexity and make calculations more straightforward.
Simplified expressions are not only easier to compute but also more intuitive to understand, leading to fewer errors and clearer mathematical communication.