Problem 160 - College Algebra

Question

In parts (a) and (b), complete each statement. a. \(\frac{b^{7}}{b^{3}}-\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}-b^{2}\) b. \(\frac{b^{8}}{b^{2}}-\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}-b^{2}\) c. Generalizing from parts (a) and (b), what should be done with the exponents when dividing exponential expressions with the same base?

Step-by-Step Solution

Answer

a. The simplified form of the given expression is \(-b^{2}\) \n b. The simplified form of the given expression is \(-b^{2}\) \n c. The general rule for dividing exponential expressions with the same base is to subtract the exponent in the denominator from the exponent in the numerator.

Step 1: Simplify the Initial Fraction

The first fraction in the expression \(b^{7}/b^{3}\) can be simplified by subtracting the exponent of the denominator from the exponent of the numerator using the quotient of powers property: \(b^{7-3}=b^{4}\). Do the same for \(b^{8}/b^{2}\) to get \(b^{8-2}=b^{6}\).

Step 2: Simplify the Next Fraction

Next, We have to simplify the fraction \(\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b \cdot b}\) for both expressions. In the numerator, we count that 'b' appears 7 times, that's \(b^{7}\), and 'b' appears 3 times in the denominator, that's \(b^{3}\). So, from the rule, we subtract to simplify, \(b^{7-3}=b^{4}\). The same operation is performed on the fraction \(\frac{b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}\), where 'b' appears 8 times in the numerator giving \(b^{8}\) and 2 times in the denominator which gives \(b^{2}\). So we get \(b^{8-2}=b^{6}\).

Step 3: Complete the Simplification

Finally, subtract \(b^{2}\) from both expressions to get \(b^{4}-b^{4}-b^{2}=0-b^{2}=-b^{2}\) for the first expression and \(b^{6}-b^{6}-b^{2}=0-b^{2}=-b^{2}\) for the second expression.

Step 4: Formulate the General Rule

The general rule for dividing exponential expressions with the same base is to subtract the exponent of the denominator from the exponent of the numerator. This rule applies irrespective of the value of the exponents as long as the base remains the same.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical expressions involving exponents. For instance, in the expression \(b^7\), "b" is the base and "7" is the exponent. This denotes multiplying the base, "b", by itself seven times. In mathematics, exponential expressions are used to simplify repeated multiplication.

With exponential expressions, a set of rules can be applied that makes complex calculations easier. By using these rules, expressions can be simplified without performing long multiplication processes. For example:

• In the expression \(b^{3}\), we have "b" multiplied by itself three times: \(b \cdot b \cdot b\).
• For \(b^{7}\), it expands to: \(b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b\).

Recognizing the power of exponential expressions helps in solving math problems more efficiently, especially when dealing with large numbers or repeating patterns.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often includes combining like terms or using mathematical properties to make an expression easier to work with. In our context, simplifying exponential expressions includes applying the rules of exponents to make the expression more manageable.

Consider the expression \(\frac{b^{7}}{b^{3}}\). By using the rules of exponents, we can simplify this expression to a cleaner form. Since both the numerator and the denominator have the same base, "b", we can subtract the exponents to simplify it to \(b^{4}\).

The goal of simplifying expressions is to make calculations less complex. This can involve reducing long expressions to shorter and more concise forms, as seen with exponential terms in mathematical operations.

Exponent Rules

Exponent rules provide a framework for working with powers in mathematical expressions. They are crucial for simplifying exponential expressions and are regularly used in various areas of mathematics. Here are some key rules:

• Quotient of Powers: If you are dividing two exponential expressions with the same base, subtract the exponents. For example, \(\frac{b^7}{b^3} = b^{7-3} = b^4\).
• Power of a Power: When raising an exponential expression to another power, multiply the exponents. For example, \((b^3)^2 = b^{3 \times 2} = b^6\).
• Product of Powers: When multiplying exponential expressions with the same base, add the exponents. For instance, \(b^2 \cdot b^3 = b^{2+3} = b^5\).

Applying these rules helps in simplifying and solving equations involving powers, making calculations more straightforward and less time-consuming. Understanding and using exponent rules effectively is a strong foundation in algebra and higher levels of mathematics.