Problem 11
Question
Write each of the following so that only positive exponents appear. $$ \frac{x^{3} y^{4}}{z^{-8}} $$
Step-by-Step Solution
Verified Answer
Question: Rewrite the given expression so that it does not contain any negative exponents: \(\frac{x^3y^4}{z^{-8}}\)
Answer: \(x^3y^4z^8\)
1Step 1: Identify negative exponents
Notice that the only negative exponent in the expression is \(z^{-8}\), which is in the denominator.
2Step 2: Convert the negative exponent to positive
To convert the negative exponent to positive, move \(z^{-8}\) from the denominator to the numerator, which will change the exponent's sign to positive:
$$
\frac{x^3 y^4 }{z^{-8}} = x^3 y^4 \cdot z^8
$$
3Step 3: Write the final answer
The expression now contains only positive exponents, so the final answer is:
$$
x^3 y^4 z^8
$$
Key Concepts
Negative ExponentsExponent RulesAlgebraic Expressions
Negative Exponents
Understanding negative exponents is essential in simplifying algebraic expressions. In essence, a negative exponent indicates the reciprocal of the base raised to the absolute value of that exponent. For example, when you come across an expression like \(z^{-8}\), it can be interpreted as \(1/z^8\).
To convert negative exponents to positive, as shown in our exercise, we can follow a simple rule: move the term with the negative exponent from the numerator to the denominator, or vice versa, and then change the sign of the exponent to positive. This is exactly what was done in the given solution: the term \(z^{-8}\) was moved from the denominator to the numerator, making the exponent positive. This technique helps in maintaining the equivalence of the expression, allowing for a clearer understanding and often a simplification of the expression.
To convert negative exponents to positive, as shown in our exercise, we can follow a simple rule: move the term with the negative exponent from the numerator to the denominator, or vice versa, and then change the sign of the exponent to positive. This is exactly what was done in the given solution: the term \(z^{-8}\) was moved from the denominator to the numerator, making the exponent positive. This technique helps in maintaining the equivalence of the expression, allowing for a clearer understanding and often a simplification of the expression.
Exponent Rules
Exponent rules, or laws of exponents, are a set of guidelines that describe how to handle various operations involving exponents. These rules make working with exponential expressions straightforward and consistent. Here are some of the key exponent rules that are beneficial to keep in mind:
- Product of Powers: To multiply two exponents with the same base, you add the exponents. For instance, \(a^m \times a^n = a^{m+n}\).
- Quotient of Powers: To divide two exponents with the same base, you subtract the exponents. An example includes \(a^m \times a^{-n} = a^{m-n}\), which is relevant to our initial problem where we moved \(z^{-8}\) to the numerator.
- Power of a Power: To raise an exponent to another power, you multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).
- Zero Exponent: Any base (except zero) raised to the power of zero is always one, as in \(a^0 = 1\).
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value or a range of values. They can include terms, constants, coefficients, and exponents, and can be as simple as \(x+5\) or as complex as the one we encountered in our exercise. When working with algebraic expressions, especially to achieve a form that only contains positive exponents, it's crucial to combine like terms, simplify operations, and apply exponent rules where necessary.
To optimize clarity and avoid potential errors, organize your work by grouping similar terms and applying operations step by step. For instance, our initial exercise presented an expression where the simplification step was focused on eliminating the negative exponent, which involved repositioning the term and ensuring all remaining exponents were positive. When expressions involve multiple variables, as in the case of \(x^3y^4z^8\), it’s essential to treat each variable independently unless they are like terms, which in this scenario, they are not. Simplifying algebraic expressions often involves making them more understandable, which ultimately can be of great help when solving equations or evaluating expressions.
To optimize clarity and avoid potential errors, organize your work by grouping similar terms and applying operations step by step. For instance, our initial exercise presented an expression where the simplification step was focused on eliminating the negative exponent, which involved repositioning the term and ensuring all remaining exponents were positive. When expressions involve multiple variables, as in the case of \(x^3y^4z^8\), it’s essential to treat each variable independently unless they are like terms, which in this scenario, they are not. Simplifying algebraic expressions often involves making them more understandable, which ultimately can be of great help when solving equations or evaluating expressions.