Problem 45 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ x^{9} y^{-6} z^{-1} w^{-5} r^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: \(x^{9} y^{-6} z^{-1} w^{-5} r^{-2}\) Answer: \(\frac{x^9}{y^6 z w^5 r^2}\)

Step 1: Identify the negative exponents

First, identify which variables have negative exponents in the given expression: $$ x^{9} y^{-6} z^{-1} w^{-5} r^{-2} $$ The variables with negative exponents are \(y^{-6}\), \(z^{-1}\), \(w^{-5}\), and \(r^{-2}\).

Step 2: Apply the exponent rule

Now, apply the exponent rule for negative exponents: $$ a^{-n} = \frac{1}{a^n} $$ Implement this rule for each variable with a negative exponent while keeping the other variables in the expression.

Step 3: Rewrite the expression with positive exponents

Rewrite the given expression using the rule for negative exponents: $$ x^{9} y^{-6} z^{-1} w^{-5} r^{-2} = x^9 \cdot \frac{1}{y^6} \cdot \frac{1}{z} \cdot \frac{1}{w^5} \cdot \frac{1}{r^2} $$

Step 4: Write the final expression

Finally, write the expression in a simplified form with only positive exponents: $$ x^9 \cdot \frac{1}{y^6} \cdot \frac{1}{z} \cdot \frac{1}{w^5} \cdot \frac{1}{r^2} = \frac{x^9}{y^6 z w^5 r^2} $$ So, the given expression with only positive exponents is: $$ \frac{x^9}{y^6 z w^5 r^2} $$

Key Concepts

Negative Exponents

Negative exponents can be a bit confusing at first, but they are simpler than they seem. When you see a negative exponent, like \( a^{-n} \), it means that the base \( a \) is on the `wrong side` of a fraction. We use the reciprocal to flip it to the other side, so \( a^{-n} \) becomes \( \frac{1}{a^n} \).
This is because a negative exponent indicates division rather than multiplication.

For example, if you have \( y^{-6} \), you can rewrite it as \( \frac{1}{y^6} \). This turns your negative exponent into a positive exponent. Remember to apply this rule to each part of the expression that has a negative exponent to achieve the desired form with all positive exponents.

Algebraic Expressions

Algebraic expressions can contain numbers, variables, and operators such as addition or multiplication. In our problem, the expression provided is \( x^{9} y^{-6} z^{-1} w^{-5} r^{-2} \).

This expression is composed of base variables \( x, y, z, w, \) and \( r \) and their associated exponents. Handling these expressions often involves manipulating variables according to certain rules.
To simplify or alter the form of an algebraic expression, knowing how to handle negative exponents and positive exponents is crucial.

When addressing these expressions, always ensure that you clearly identify and properly manipulate the exponents as instructed, so every part of your final algebraic expression is expressed with positive exponents.

Exponent Rules

Exponent rules are a set of guidelines that tell us how to handle powers and the numbers or variables they are applied to. Two important rules to note are the rule for multiplying expressions with the same base and the rule for negative exponents.

• Multiplying like bases means adding their exponents, such as \( a^m \times a^n = a^{m+n} \).
• For negative exponents, remember \( a^{-n} = \frac{1}{a^n} \), which helps convert any negative exponent into a positive one.

In our exercise, we used the negative exponent rule to transform the expression \( x^{9} y^{-6} z^{-1} w^{-5} r^{-2} \) into a form \( \frac{x^9}{y^6 z w^5 r^2} \) where all exponents are positive.
Following these rules ensures that your expressions are not only simplified but also in accordance with mathematical conventions.