Problem 39 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \frac{7 x}{y^{-3} z^{-2}} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: \(\frac{7x}{y^{-3}z^{-2}}\) Answer: \(7xy^3z^2\)

Step 1: Simplify the expression with negative exponents to positive exponents

Recall the rule: \(a^{-n} = \frac{1}{a^n}\). We will apply this rule to rewrite the negative exponents as positive exponents. For the given expression: $$ \frac{7x}{y^{-3}z^{-2}} $$ we have negative exponents on the \(y\) and \(z\) terms. Applying the rule, this expression can be rewritten as $$ \frac{7x}{\frac{1}{y^3} \cdot \frac{1}{z^2}} $$

Step 2: Multiply across the fractions

To finalize the expression with positive exponents, we will multiply across the fractions: $$ \frac{7x}{1} \cdot y^3 \cdot z^2 = 7xy^3z^2 $$ Now the expression has only positive exponents. The final expression is: $$ 7xy^3z^2 $$

Key Concepts

Negative Exponents

Negative exponents can often feel tricky, but they are straightforward once you understand the basic rule. A negative exponent means you take the reciprocal of the base raised to the opposite, positive exponent. In simpler terms, if you have \( a^{-n} \), it is equivalent to \( \frac{1}{a^n} \).

This idea flips the base to the denominator if it is in the numerator and vice versa. For example, if you see \( y^{-3} \), it becomes \( \frac{1}{y^3} \). This transformation is crucial when working to simplify expressions because it allows us to rewrite expressions using only positive exponents.

Using this concept of negative exponents makes it easier to perform operations on algebraic expressions and understand the behavior of exponential terms in equations. Always remember to convert negative exponents to positives before proceeding further in calculations.

Exponent Rules

Exponent rules are essential when you are dealing with simplifications and calculations involving powers. Let's start with some basic exponent rules which include:

• Power of a power: \((a^m)^n = a^{m \times n}\)
• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\) if \(aeq0\).

These rules are useful for manipulating expressions and answering algebra questions efficiently. They help in converting expressions with negative exponents into positive ones by shifting the power (as seen in the original exercise).

When you see a term like \( y^{-3} \) in a denominator, applying the rule means you take \( y^3 \) and move it to the numerator, turning it positive. Multiplying fractions and ensuring expressions contain only positive exponents become manageable when these rules are laid out clearly. Practicing these rules will make solving algebraic expressions less daunting.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, multiplication, and exponentiation. They form the backbone of algebra. Understanding how to simplify, manipulate, and rewrite these expressions is key to mastering algebra.

An expression such as \( \frac{7x}{y^{-3}z^{-2}} \) involves variables with negative exponents. By applying exponent rules, the expression can be rewritten to \( 7xy^3z^2 \) with only positive exponents, making it simpler to understand or use in solving equations.

Working with algebraic expressions often involves simplifying them or solving them for one variable. This process includes removing negative exponents and applying exponent rules.

UIKey tasks include:

• Recognizing the parts of an expression
• Understanding the operations involved
• Applying the correct rules to simplify or transform the expression

By mastering these skills, you will find yourself better equipped to tackle complex algebraic problems.