Problem 65 - 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. $$ \frac{e^{11}}{e^{11}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{e^{11}}{e^{11}}$ using the quotient rule of exponents. Answer: 1

Step 1: Understand the quotient rule of exponents

The quotient rule of exponents is a basic exponent rule that states $$ \frac{a^m}{a^n} = a^{m-n}, $$ for any nonzero base a and whole numbers m and n. This rule implies that when we are dividing two expressions with the same base and whole-number exponents, we can simply subtract the exponent of the denominator from the exponent of the numerator.

Step 2: Apply the quotient rule to the expression

Given the expression: $$ \frac{e^{11}}{e^{11}} $$ Using the quotient rule of exponents, we can write $$ \frac{e^{11}}{e^{11}} = e^{11-11} $$

Step 3: Calculate the new exponent

Now, we can subtract the exponents: $$ 11 - 11 = 0 $$ So the expression becomes: $$ e^{11-11} = e^0 $$

Step 4: Simplify the exponent equals to zero

Recall that any nonzero number raised to the power of 0 is equal to 1 (by definition). So we have: $$ e^0 = 1 $$

Step 5: Write the final simplified expression

The simplified expression after applying the quotient rule of exponents is: $$ \frac{e^{11}}{e^{11}} = 1 $$

Key Concepts

product rule of exponents

The product rule of exponents is a fundamental concept when working with exponential expressions. It tells us how to handle multiplying similar bases raised to different powers.

When you have an expression like \(a^m \times a^n\), the product rule states that you simply add the exponents:

• Formula: \(a^m \times a^n = a^{m+n}\)
• This rule only works when the bases are the same nonzero whole number.

For example, if you have \(3^2 \times 3^4\), applying the rule results in \(3^{2+4} = 3^6\).

Understanding and using the product rule effectively helps in simplifying complex expressions by condensing them into a simpler, single-exponent format.

simplifying expressions

Simplifying expressions involves reducing a mathematical expression to its simplest form. This often involves using rules and properties like those of exponents.

To simplify, you'll need to:

• Apply exponent rules like the product, quotient, and power rules.
• Combine like terms.
• Follow order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

For instance, simplifying \(\frac{e^{11}}{e^{11}}\) using exponent rules shows how subtraction of exponents and simplifying \(e^0\) leads to the simplest form, which is 1.

The main goal is to reduce the expression while maintaining its original value. This makes it easier to work with in further calculations.

whole numbers

Whole numbers are the set of numbers without fractions or decimals. They include all positive integers along with zero: 0, 1, 2, 3, and so on.

Important points about whole numbers:

• They don’t include negatives or fractions.
• The smallest whole number is 0.
• They are used frequently as exponents due to their simplicity.

In relation to exponent rules, whole numbers simplify calculations because their properties are consistent and straightforward. Using the quotient rule, for example, is simpler when dealing with whole numbers, since you are only subtracting whole numbers rather than working with fractions or decimals.

basic exponent rules

Basic exponent rules are crucial for simplifying mathematical expressions involving exponents. These rules include:

• **Product Rule**: As mentioned earlier, multiply similar bases and add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Quotient Rule**: For division: \( \frac{a^m}{a^n} = a^{m-n}\), if the bases are the same.
• **Power Rule**: If raising a power to another power: \((a^m)^n = a^{m \times n}\).
• **Zero Exponent Rule**: Any base except zero raised to the power of zero is 1: \(a^0 = 1\).

These rules are foundational for more complex mathematical operations and help in efficiently breaking down and understanding larger problems. By mastering these, you make more substantial progress in learning algebra and other branches of mathematics.