Problem 69 - 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. $$ \frac{8 a^{4} b^{0}}{4 a^{3}} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $2a$.

Step 1: Simplify the numerical coefficients

First, we'll simplify the numerical coefficients by dividing 8 by 4. $$ \frac{8}{4} = 2 $$

Step 2: Apply the quotient rule to the variable a

Next, we'll apply the quotient rule to the variable a. Recall that the quotient rule states that \(\frac{a^{m}}{a^{n}} = a^{m-n}\). So we get: $$ \frac{a^{4}}{a^{3}} = a^{4-3} = a^1 $$

Step 3: Simplify the variable b with exponent 0

Now, let's address the b term. As b is raised to the power of 0, we'll use the rule \(b^0=1\). So, \(b^0\) simplifies to 1.

Step 4: Write the simplified expression

Now that we have simplified each part of the expression, we can write the simplified expression as a product: $$ 2a^1 \cdot 1 $$ However, as multiplying by 1 does not change the value, our final simplified expression is: $$ 2a $$ So, the simplified expression for the given problem is \(2a\).

Key Concepts

Product Rule of Exponents

The product rule of exponents is a key concept in algebra that helps simplify expressions involving powers. When we multiply two expressions with the same base, we can add the exponents together. In mathematical terms, if you have two expressions like \(a^m\) and \(a^n\), the product rule says \(a^m \cdot a^n = a^{m+n}\).

This rule greatly simplifies calculations and is particularly useful when dealing with long expressions or algebraic fractions. It's important to ensure that the bases are the same, as this allows the exponents to be directly combined. Understanding this rule makes it easier to perform operations on algebraic expressions and can help solve more complex equations.

Quotient Rule of Exponents

The quotient rule of exponents is similar to the product rule but applies to division instead of multiplication. When you divide two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This can be expressed as \(\frac{a^m}{a^n} = a^{m-n}\).

This rule is incredibly useful when simplifying algebraic fractions, as seen in the exercise. For the term \(\frac{a^4}{a^3}\), applying the quotient rule means subtracting the exponents. This simplification results in \(a^{4-3} = a^1\) or simply \(a\).

Carefully applying this rule allows us to reduce fractions to their simplest form and manage expressions more effectively.

Exponents

Exponents are a fundamental aspect of mathematics, especially algebra. They are shorthand notation for repeated multiplication. For example, \(a^3\) means \(a \times a \times a\). The number \(a\) is called the base, and the number 3 is the exponent or power, which indicates how many times the base is used as a factor.

Exponents make it easier to write and work with very large or small numbers. They follow specific rules, such as the product and quotient rules, which simplify expressions. Another important property is that any nonzero base raised to the power of zero is 1: \(a^0 = 1\).

Understanding these concepts helps in manipulating and simplifying expressions, essential skills in algebra.

Algebraic Fractions

Algebraic fractions, much like numerical fractions, involve division but include variables in the numerator, the denominator, or both. Simplifying these fractions often involves using exponent rules to transform and reduce them.

In the given problem, \(\frac{8a^4b^0}{4a^3}\), the expression involves both numerical coefficients and variables with exponents. By simplifying these components using the product and quotient rules, we can reduce it to a more manageable form. The final simplified expression, \(2a\), results from dividing numerical coefficients and applying exponent rules to the variables.

Mastering algebraic fractions is crucial for solving equations and understanding more advanced mathematical concepts.