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Chapter 7: Problem 41

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

Short Answer

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

Step by step solution

01

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: \[ x \times \frac{x}{1} \]
02

Simplify the Multiplication

Simplify the multiplication by multiplying the numerators and the denominators: \[ x \times x = x^2 \]
03

Final Simplified Form

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } T \text { are constant; the relationship of } R \text { and } C \text {. } $$

For exercises 1-10, (a) solve. (b) check. $$ \frac{3}{5} x-\frac{1}{4}=\frac{9}{10} $$

In radiography, a grid reduces the effect of X-ray scattering. The relationship of the interspace distance on the grid, \(x\), and the grid ratio, \(y\), is an inverse variation. When the interspace distance on a grid is 300 micrometers, the grid ratio is 8 . Write an equation that represents this variation. Include the units.

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ C, T, \text { and } F \text { are constant; the relationship of } P_{i} \text { and } P_{m} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=4, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=10\).
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