Problem 62 - 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. $$ \frac{m^{17} n^{12}}{m^{16} n^{10}} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: \(\frac{m^{17} n^{12}}{m^{16} n^{10}}\) Answer: \(mn^2\)

Step 1: Apply the quotient rule for exponents

First, we apply the quotient rule for each base in the expression: $$ \frac{m^{17} n^{12}}{m^{16} n^{10}} = m^{17-16} \cdot n^{12-10} $$

Step 2: Simplify the expressions

Now, we simply subtract the exponents to find the simplified form: $$ m^{17-16} \cdot n^{12-10} = m^{1} \cdot n^{2} $$

Step 3: Write the final simplified expression

The final simplified expression, after applying the quotient rule for exponents, is: $$ m^1 \cdot n^2 = mn^2 $$ In conclusion, the simplified expression for the given problem is \(mn^2\).

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is like tidying up a messy room into an organized space. The goal is to make complex or cluttered algebraic expressions easier to understand and work with by using rules of arithmetic and algebra.

When we simplify expressions that involve exponents, it's important to remember that simplification is all about reducing the complexity while maintaining equivalence. For example, the expression \frac{m^{17} n^{12}}{m^{16} n^{10}}\ can appear daunting at first glance. To simplify this, we start by breaking down the expression for each base (m and n) separately, applying exponent rules to simplify each portion. In the step-by-step solution provided, this is exactly what was done using the quotient rule for exponents, which helps in minimizing the expression to its core, without altering its value.

Remember the End Goal

By simplifying, we're not changing the expression's value but providing a more efficient way of expressing the same mathematical relationship. The result of \frac{m^{17} n^{12}}{m^{16} n^{10}}\ as \( mn^2 \) represents a tidy and concise form, showing that simplifying can immensely improve readability and ease of use in further calculations.

Product Rule of Exponents

Understanding the product rule of exponents is like learning a simple shortcut in math. The product rule states that when multiplying two exponents with the same base, you can simply add the exponents together while keeping the base the same. The mathematical notation for this rule is: \forall a eq 0\, \( a^m \times a^n = a^{m+n} \)

This rule is essential when working with simplifying expressions, especially when they involve products of powers. It's important to note that this rule is only applicable when the bases are identical. When faced with an expression like \( m^{17} \times m^{16} \), for example, the product rule guides us to add the exponents because the base (m) remains constant. The result would therefore be \( m^{17+16} \). However, in our original exercise, we are working with the quotient of powers, not the product, and thus the product rule doesn't directly apply.

Expanding Our Toolset

Beyond simplifying, the product rule can also be helpful in expanding expressions or in problems that require manipulating powers. It's another valuable 'mathematical power tool' in your algebra toolbox, ensuring that exponent-laden expressions do not slow you down.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, the base, to the power of an exponent. In exponentiation, the exponent indicates how many times to multiply the base by itself. For instance, \( 2^3 \) means you multiply 2 by itself three times (\( 2 \times 2 \times 2 = 8 \)).

When dealing with exponentiation in algebra, it's not just numbers we work with; variables can also act as bases and exponents. This allows exponentiation to express very large or very small quantities and to handle growth or decay processes in disciplines like physics, economics, and biology.

Key Properties

Some key properties of exponentiation, used in simplifying expressions, include the product rule (as previously discussed), the quotient rule (when dividing like bases, we subtract the exponents), and the power rule (when raising an exponent to another power, we multiply the exponents). The correct application of these rules is crucial to working efficiently with exponential expressions.

Revisiting our original exercise, the expression \(m^{17-16} \times n^{12-10} = m^1 \times n^2\) after applying the quotient rule, highlights how exponentiation was crucial in achieving a simplified form. Always ensure that the base and its exponent are treated as a single unit when applying any of the rules of exponentiation.