Problem 41 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ x \div \frac{1}{x} $$

Step-by-Step Solution

Answer

The simplified form is \( x^2 \).

Step 1: Rewrite the Division

Rewrite the division operation as a multiplication by the reciprocal. In this case, dividing by a fraction is the same as multiplying by its reciprocal: \[ x \times \frac{x}{1} \]

Step 2: Simplify the Multiplication

Simplify the multiplication by multiplying the numerators and the denominators: \[ x \times x = x^2 \]

Step 3: Final Simplified Form

The final simplified form of the expression is: \[ x^2 \]

Key Concepts

Reciprocal

Understanding the concept of a reciprocal is fundamental in simplifying algebraic expressions. The reciprocal of a number is essentially 'flipping' the number's fraction. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). If you are working with whole numbers, you can consider them as having a denominator of 1. Thus, the reciprocal of x is \(\frac{1}{x}\). This is crucial because division by a fraction can be rewritten as multiplication by its reciprocal. For instance, dividing x by \(\frac{1}{x}\) becomes multiplying x by \(\frac{x}{1}\), simplifying the steps significantly.

Multiplication

Multiplication is one of the basic arithmetic operations that you often use in algebra. In the process of simplifying expressions, multiplication plays a key role, especially when it comes to dealing with fractions and their reciprocals. When you multiply two numbers, you essentially add their exponents if they have the same base. In the exercise, multiplying x by its reciprocal \(\frac{x}{1}\) involves straightforward multiplication of the terms \(\frac{x}{1} x\) yielding \(x \times x = x^2\). The multiplication is simplified as each term is multiplied directly to give the product.

Simplification

Simplification in algebra means making an expression easier to work with or understand, often by reducing it to its most basic form. This involves performing all possible arithmetic operations, combining like terms, and reducing fractions if possible. In our exercise example, the initial problem \(x \div \frac{1}{x}\) can be simplified by converting the division into multiplication by the reciprocal. Converting and multiplying the terms simplifies the expression to its lowest basic form \(x^2\). Simplification helps in making complex expressions manageable and is useful in solving more complicated algebraic equations.

Exponents

Exponents indicate how many times a number (the base) is multiplied by itself. Understanding exponents helps you manage repeated multiplication in algebra. For example, in our exercise, when we multiply \(x \times x\), we get \(x^2\). This is because we are multiplying x by itself once. When you come across expressions involving exponents, remember these basic rules: \(a^m \times a^n = a^{m+n}\) and \( (a^m)^n = a^{mn} \). Exponents simplify expressions and make it easier to work with large numbers succinctly in algebra.