Problem 42 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

\( z \div \frac{1}{z} = z^2 \).

Step 1: Identify the Division

Recognize that the exercise involves dividing a variable by a fraction. The expression to simplify is \( z \div \frac{1}{z} \) .

Step 2: Rewrite the Division as Multiplication

Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, rewrite the division \( z \div \frac{1}{z} \) as a multiplication: \( z \times z \).

Step 3: Simplify the Multiplication

Now, multiply the terms together: \( z \times z \). The product of the same base is given by adding the exponents. Thus, \( z \times z = z^2 \).

Key Concepts

multiplying variables

In algebra, multiplying variables is straightforward if you understand the basics. When you multiply two variables together, you're essentially combining them. For example, if you have two variables, say 'a' and 'b', multiplying them results in 'ab'. If the variables are the same, like 'z' and 'z', you can represent them as \(z \times z\). This concept is fundamental when dealing with algebraic expressions and sets the stage for further simplification.

reciprocal of a fraction

A reciprocal is created by flipping the fraction. For instance, the reciprocal of \(\frac{1}{z}\) is \(z\). This is useful in algebra for simplifying expressions. Instead of dividing by a fraction, you multiply by its reciprocal. In our example, dividing \(z\) by \(\frac{1}{z}\) is the same as multiplying \(z\) by its reciprocal, \(z\). This step simplifies complex division problems into easier multiplication tasks, easing the path to further simplification.

algebraic simplification

Algebraic simplification is the process of reducing an algebraic expression into its simplest form. This often involves combining like terms, canceling out terms, or applying basic arithmetic operations. In our example, transforming \(z \times z\) into \(z^2\) is a perfect instance. Simplification makes expressions easier to understand and solve, and it's a crucial skill in mathematics. With practice, algebraic simplification becomes second nature and helps lay a solid foundation for solving more complex problems.

exponents in multiplication

When you multiply variables with exponents, you add the exponents if the bases are the same. For instance, if \(a^m\) and \(a^n\) are multiplied, it becomes \(a^{m+n}\). This rule simplifies expressions involving the same base. In the example of \(z \times z\), both terms have an implied exponent of 1. Adding the exponents \(1+1\) results in \(z^2\). Understanding exponents in multiplication is key to mastering algebra as it helps simplify and solve equations efficiently.