Problem 49 - 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. $$ \frac{8^{5}}{8^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{8^{5}}{8^{3}}\). Answer: 64

Step 1: Identify the bases and exponents

In the given expression, we have \(\frac{8^{5}}{8^{3}}\). Here, the base is 8, while the exponents are 5 and 3 for the numerator and denominator, respectively.

Step 2: Apply the quotient rule of exponents

The quotient rule of exponents states that for an expression in the form \(\frac{a^{m}}{a^{n}}\), the result is \(a^{m-n}\). Apply this rule to the given expression: $$ \frac{8^{5}}{8^{3}} = 8^{5-3} $$

Step 3: Subtract the exponents

Perform the subtraction in the exponent: $$ 8^{5-3} = 8^{2} $$

Step 4: Calculate the result

Now, calculate the value of the expression: $$ 8^{2} = 8 \times 8 = 64 $$ Therefore, the simplified expression is 64.

Key Concepts

Product Rule of Exponents

Understanding the product rule of exponents is essential for multiplying expressions with the same base. When you multiply powers with the same base, you simply add the exponents. This rule is expressed as: \( a^m \times a^n = a^{m+n} \). This concept helps to simplify expressions and solve complex equations more easily. It’s especially useful in algebra where terms with the same base are common. Here are a few simple pointers on how to apply the product rule:

• Identify that both expressions have the same base.
• Add the exponents of the terms.
• Rewrite the expression with the summed exponent.

By mastering the product rule, you can swiftly handle multiplication of exponential expressions without multiplying each individual term, making problem-solving quicker and more efficient.

Simplifying Expressions

Simplifying expressions involves transforming them into their simplest form, where they can be easily understood or calculated. In the context of exponents, this process often involves rules like the quotient and product rules to reduce complexity. When simplifying expressions with exponents:

• Use the quotient rule when dividing expressions with the same base.
• Apply the product rule when multiplying expressions with the same base.
• Simplify the exponents before doing any arithmetic operations.

Through simplifying expressions, you make them shorter, easier to manage, and more helpful in uncovering relationships between variables in algebraic equations. The main goal is to keep the expression as minimalistic as possible while maintaining equivalency, which aids in both understanding and solving the expressions.

Whole Number Exponents

Whole number exponents are exponents that are positive integers, including zero. When dealing with exponents, it's crucial to understand how multiplication works to relate base numbers through powers. A whole number exponent indicates how many times you multiply the base by itself. For instance, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \), which equals 81.Key characteristics of whole number exponents include:

• They are always positive integers or zero.
• Reflect repeated multiplication of a base.
• They simplify expressions through exponential rules like the product or quotient rule because they’re straightforward to manage.

Understanding whole number exponents empowers you to effectively break down and tackle mathematical expressions and problems involving exponential growth or repeated multiplications.