Problem 31 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{1}{x^{8}} \cdot x^{5} $$

Step-by-Step Solution

Answer

The simplified form of the given term is \(\frac{1}{x^3}\)

Step 1: Apply the rule of exponents for multiplication

When you multiply two exponents with the same base, like this case where the base is x, you simply add the exponents. Thus, we have \(x^{-8} \cdot x^{5} = x^{-8+5}\)

Step 2: Simplify the exponent

Adding the exponents -8 and 5, we get -3. Therefore, the simplification of the given term is \(x^{-3}\)

Step 3: Rewrite the exponent

The term \(x^{-3}\) can be written as \(\frac{1}{x^3}\), as the negative exponent represents a reciprocal. Therefore the final simplified form of the given term is \(\frac{1}{x^3}\)

Key Concepts

Rules of Exponents

Understanding the rules of exponents is essential for simplifying expressions involving powers. Exponents, also known as powers, are a shorthand way to express repeated multiplication of the same number. The most basic rules include the product rule, power rule, and quotient rule.

The product rule states that when multiplying two expressions with the same base, you add the exponents; for example, \(a^m \times a^n = a^{m+n}\). The power rule indicates that when you raise an exponent to another power, you multiply the exponents; for example, \((a^m)^n = a^{m \times n}\). The quotient rule provides guidance when dividing exponents with the same base: subtract the exponent in the denominator from the exponent in the numerator; for example, \(a^m \big/ a^n = a^{m-n}\), as long as \(a\) is not zero.

These rules are the foundation for simplifying complex algebraic expressions and are applicable for both positive and negative integer exponents.

Exponent Multiplication

When working with exponents, multiplying terms with the same base involves a straightforward process. According to the rule of exponent multiplication, you maintain the base and sum up the exponents. This approach assumes the exponents are integers and does not require the base to be changed.

For instance, \(x^a \times x^b = x^{a+b}\). The operation of adding the exponents effectively counts the total number of times the base is multiplied by itself. This understanding helps in simplifying complicated expressions and is an essential algebraic skill.

Example of Exponent Multiplication

If you have \(x^3 \times x^4\), applying the rule of exponent multiplication results in \(x^{3+4} = x^7\), since the base \(x\) is the same for both terms.

Negative Exponents

Negative exponents can initially be confusing, but they represent an important mathematical concept. A negative exponent indicates that the term is the reciprocal of the positive exponent. Specifically, \(a^{-n} = \frac{1}{a^n}\) where \(a\) is nonzero. This is known as the reciprocal rule.

The reciprocal rule enables the transformation of negative exponents into their positive counterparts by reciprocation. It is imperative to understand that a negative exponent does not make the value negative; it simply moves the term into the denominator of a fraction.

Application of Negative Exponents

An example would be \(x^{-2}\), which we can write as \(\frac{1}{x^2}\), demonstrating how negative exponents are used to denote division by that base raised to the positive exponent.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They are essentially the building blocks of algebra and are used to represent real-world problems in abstract mathematical terms.

Understanding how to manipulate these expressions using algebraic rules is key to solving equations and inequalities. These manipulations include operations like expansion, factoring, and simplifying using the rules of exponents discussed earlier. Mastery of algebraic expressions allows one to dissect complex problems and find solutions methodically.

It's also important to differentiate between an algebraic expression and an equation. An algebraic expression doesn't include an equality sign and therefore can't be solved, only simplified. An equation, on the other hand, does have an equality sign and can be solved for a variable.