Problem 38 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ x^{3} y^{2} z^{-6} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(x^{3}y^{2}z^{-6}\) using only positive exponents. Answer: \(\frac{x^{3}y^{2}}{z^6}\)

Step 1: Identify terms with negative exponents

In the given expression, \(x^{3}y^{2}z^{-6}\), the term with a negative exponent is \(z^{-6}\).

Step 2: Apply the exponent rule to the term with a negative exponent

According to the exponent rule \(a^{-m} = \frac{1}{a^m}\), we can rewrite the term \(z^{-6}\) as \(\frac{1}{z^6}\).

Step 3: Substitute the new expression

Replace the term \(z^{-6}\) in the given expression with the rewritten expression from Step 2: $$ x^{3}y^{2}\left(\frac{1}{z^6}\right) $$

Step 4: Simplify the expression

The rewritten expression \(x^{3}y^{2}\left(\frac{1}{z^6}\right)\) can be simplified to: $$ \frac{x^{3}y^{2}}{z^6} $$ So the expression \(x^{3} y^{2} z^{-6}\) with only positive exponents is: $$ \frac{x^{3}y^{2}}{z^6} $$

Key Concepts

Negative Exponents

Negative exponents can sometimes be confusing, but they follow a simple rule. When you see a negative exponent, think of it as an instruction to "flip" the base to the other side of a fraction. This is known as the "reciprocal" rule. For example, if you have a number or variable with a negative exponent, like \( a^{-m} \), it can be rewritten as \( \frac{1}{a^m} \).
This means instead of counting how many times to multiply the base into itself, you are determining how many times to divide 1 by the base. In the exercise, the expression \( z^{-6} \) becomes \( \frac{1}{z^6} \).
Understanding this helps to easily manage and simplify expressions by transforming negative exponents into positive ones, making expressions clearer and easier to work with.

Positive Exponents

Positive exponents are more straightforward to understand. They tell you how many times to multiply the base by itself. For instance, \( x^3 \) means \( x \) is multiplied by itself three times: \( x \times x \times x \).
When dealing with expressions, always aim to convert any negative exponents to positive ones, as it simplifies the expression and makes it more standard in algebraic solutions.
This conversion process allows us to work with a more manageable form, which is why the exercise emphasizes writing expressions with only positive exponents. In the original exercise, the terms \( x^3 \) and \( y^2 \) are already in their positive exponent forms and require no further adjustment.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. Understanding how to manipulate them is essential in algebra.
These expressions can include both positive and negative exponents, as well as variables that must adhere to certain rules, like non-zero restrictions. Simplifying expressions involves rewriting them using these rules to achieve a form that is easier to interpret and use in equations.
An algebraic expression can be something as simple as \( x^3y^2 \) or as complex as \( \frac{x^3y^2}{z^6} \).
Breaking down and rewriting expressions with negative exponents, as shown in the exercise, helps in simplifying algebraic work and ensuring we follow standard mathematical practices.