Problem 2 - Elementary Algebra

Question

For Problems \(1-24\), divide the monomials. $$ \frac{x^{12}}{x^{5}} $$

Step-by-Step Solution

Answer

\(x^7\)

Step 1: Identify the Base and Exponents

In the given expression \(\frac{x^{12}}{x^{5}}\), both the numerator and the denominator share the same base \(x\). This allows us to apply exponent rules to simplify.

Step 2: Apply the Quotient of Powers Rule

The Quotient of Powers rule states that for any nonzero number \(a\), \(\frac{a^m}{a^n} = a^{m-n}\). In this case, \(m = 12\) and \(n = 5\), so we can simplify the expression to \(x^{12-5}\).

Step 3: Calculate the New Exponent

Subtract the exponents: \(12 - 5 = 7\). This means the expression simplifies to \(x^7\).

Key Concepts

Quotient of Powers Rule

When dividing monomials with the same base, you need to apply the Quotient of Powers rule. This rule is a fundamental part of exponent rules that helps simplify expressions involving powers.

• The rule states that if you have the same base being divided, you subtract the exponent in the denominator from the exponent in the numerator.
• For example, with the expression \(\frac{x^{12}}{x^{5}}\), both terms have the same base \(x\).
• According to the Quotient of Powers rule, you simply subtract the exponents: \(12 - 5\).
• This means you will simplify the expression to \(x^{12-5} = x^7\).

Understanding and using the Quotient of Powers rule can greatly simplify working with polynomial expressions and is essential in solving algebraic problems efficiently.

Exponent Rules

Exponent rules are important when working with expressions involving powers, as they provide a systematic way to simplify calculations.

• The basic rules include the Product of Powers, Power of a Power, and Quotient of Powers, which we've just discussed.
• The Product of Powers rule says that when multiplying two powers with the same base, you add the exponents (\(a^m \cdot a^n = a^{m+n}\)).
• The Power of a Power rule means if you raise a power to another power, you multiply the exponents (\((a^m)^n = a^{m \cdot n}\)).
• Knowing these rules helps in simplifying expressions and solving problems quickly and accurately.

Exponent rules, like the Quotient of Powers, are tools that make dealing with large or complex expressions easier and are frequently used across different areas of mathematics and science.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. This includes using rules we learned, such as the Quotient of Powers.

• Simplifying helps make expressions easier to manage and understand.
• You often start by identifying common bases or like terms, as seen in the problem \(\frac{x^{12}}{x^{5}}\).
• Apply the relevant exponent rules, such as subtraction for division of similar bases, which we used in simplifying \(x^{12 - 5}\) to \(x^7\).
• Always double-check your results to ensure no further simplification is possible, which might involve additional rules like combining like terms or simplifying coefficients.

Efficiently simplifying mathematical expressions is crucial in making calculations more straightforward, ensuring accurate results, and preparing expressions for further calculations or evaluations.