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Chapter 1: Problem 38

Perform the multiplication or division and simplify. $$\frac{x}{y / z}$$

Short Answer

Expert verified
\( \frac{xz}{y} \) is the simplified form of the expression.

Step by step solution

01

Understand the Expression

The expression given is a fraction: \( \frac{x}{y/z} \). It involves division in the denominator, which simplifies to multiplying the numerator by the reciprocal of the denominator.
02

Apply the Rule of Division of Fractions

When dividing by a fraction \( y/z \), you multiply by its reciprocal. The reciprocal of \( y/z \) is \( z/y \). Thus, the expression becomes \( x \times \frac{z}{y} \).
03

Perform the Multiplication

Now, multiply the numerator \( x \) by the reciprocal \( \frac{z}{y} \). This results in: \( \frac{xz}{y} \).
04

Simplify if Possible

Check if \( xz \) and \( y \) have any common factors that can be simplified. If there are no common factors, this is the simplest form.

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Key Concepts

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

Fractions
Fractions represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number). In algebra, fractions can involve variables in place of numbers. They are an essential concept to understand as they form the basis for many operations, especially in algebraic expressions.
To work effectively with fractions:
  • Identify the numerator and denominator clearly.
  • Perform operations like addition, subtraction, multiplication, or division according to rules specific to fractions.
  • Understand that division by a fraction is the same as multiplication by its reciprocal.
Fractions often require simplification to express them in their simplest form. This is particularly important in algebra to ensure clarity and simplify further operations.
Reciprocal
The reciprocal of a fraction plays a crucial role in dividing fractions. The reciprocal is obtained by flipping the numerator and the denominator of the fraction. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
The reciprocal is useful because:
  • It transforms division into multiplication, which is often easier to compute.
  • It helps in simplifying expressions involving complex fractions.
When dividing fractions, like \( \frac{x}{y/z} \), you can easily convert division into multiplication by using the reciprocal of the divisor, turning the expression into multiplication, which is more straightforward to handle in algebraic expressions.
Simplification of Expressions
Simplification is the process of reducing an expression to its simplest form. It involves removing any unnecessary complexity without changing the expression's value. When simplifying, always look for opportunities to factor or cancel out common terms.
Key steps to simplify expressions include:
  • Identify common factors in the numerator and denominator and cancel them out.
  • Use algebraic identities and properties to transform expressions.
  • In the context of fractions, simplifying may involve factoring polynomials in the numerator and denominator.
Simplification makes expressions easier to interpret and handle, especially in equations or when performing additional operations such as solving or graphing. It's a vital skill in algebra that reduces errors and assists in obtaining solutions in their cleanest form.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$\left(x+\left(2+x^{2}\right)\right)\left(x-\left(2+x^{2}\right)\right)$$

Factor the expression completely. $$2 x^{3}+4 x^{2}+x+2$$

Perform the indicated operations and simplify. $$\left(x^{1 / 2}+y^{1 / 2}\right)\left(x^{1 / 2}-y^{1 / 2}\right)$$

Use the Difference of Squares Formula to factor \(17^{2}-16^{2}\). Notice that it is easy to calculate the factored form in your head but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) \(528^{2}-527^{2}\) (b) \(122^{2}-120^{2}\) (c) \(1020^{2}-1010^{2}\) Now use the Special Product Formula \((A+B)(A-B)=A^{2}-B^{2}\) to evaluate these products in your head: (d) \(79 \cdot 51\) (e) \(998 \cdot 1002\)

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$\left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2}$$
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