Problem 59 - 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. $$ \frac{m^{16}}{m^{9}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{m^{16}}{m^{9}}$ using the product rule and quotient rule of exponents. Answer: $m^7$

Step 1: The given expression is $$ \frac{m^{16}}{m^{9}} $$ As there is the same base with whole number exponents in both numerator and denominator, the quotient rule of exponents will be applied to simplify the expression. #Step 2: Apply the quotient rule of exponents#

The quotient rule of exponents states that for any nonzero base 'a' and whole number exponents 'm' and 'n', $$ a^m \div a^n = a^{m-n} $$ Applying the quotient rule to the given expression, we have $$ \frac{m^{16}}{m^{9}} = m^{16-9} $$ #Step 3: Calculate the new exponent#

Step 2: Subtract the exponent in the denominator from the exponent in the numerator: $$ 16 - 9 = 7 $$ #Step 4: Write the final simplified expression#

Substitute the new exponent in the expression: $$ m^{16-9} = m^7 $$ Therefore, the simplified expression is $$ m^7 $$

Key Concepts

Product Rule of Exponents

When simplifying expressions with exponents, the product rule is one of the first rules to learn. It is straightforward yet powerful. The product rule states that when you are multiplying two exponents with the same base, you simply add the exponents together. This rule can be mathematically expressed as:

• \( a^m \times a^n = a^{m+n} \)

Here, \( a \) is the base that must be the same for both terms, while \( m \) and \( n \) are the exponents that need to be added. For example, if you have \( x^3 \times x^4 \), applying the product rule gives you \( x^{3+4} = x^7 \). This rule simplifies expressions by reducing the number of terms.

Quotient Rule of Exponents

The quotient rule is a close relative to the product rule and is just as essential for simplifying expressions. When you divide two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator. The mathematical representation of this principle is:

• \( \frac{a^m}{a^n} = a^{m-n} \)

This rule is applicable as long as the base (\( a \)) is non-zero. In the exercise you provided, \( \frac{m^{16}}{m^9} \) simplifies by subtracting 9 from 16, resulting in \( m^{16-9} = m^7 \). It shows how effective the quotient rule is in reducing the complexity of expressions.

Simplifying Expressions

Simplifying expressions is about reducing complexity while maintaining the original value. It often involves using a mix of rules including the product and quotient rules. For example, if you start with a complex expression like \( \frac{k^5 \times k^3}{k^2} \), you can bring it down to \( k^{5+3-2} = k^6 \), efficiently minimizing the terms.
When simplifying, it is crucial to ensure all rules are applied correctly to preserve the expression's value. This process often looks like peeling layers off an onion until the simplest form is achieved. Remember to always work systematically, step by step, to avoid mistakes.

Whole Number Exponents

Exponents indicate how many times a base is multiplied by itself, and a whole number exponent is simply an exponent that is a non-negative integer. This makes calculations more straightforward.

• For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 \), which equals 81.
• The higher the exponent, the larger the result, as it involves repeated multiplication.

Understanding whole number exponents is crucial as it lays the groundwork for handling more complex operations with exponents like the ones encountered in algebra practices. Unlike fractional or negative exponents, whole numbers are simpler and often easier to work with, making them perfect for mastering the basics of exponent rules.