Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\frac{m^{4}}{n^{0}}\)

Step-by-Step Solution

Answer

The simplified expression is \(m^4\).

Step 1: Understanding Exponents

Recall the property of exponents which states that any number raised to the power of zero is equal to one: \(a^0 = 1\), where \(a\) is any non-zero number.

Step 2: Simplifying the Denominator

We see that the denominator of the expression is \(n^0\). Using the zero exponent rule, we can simplify this to \(1\). So, the expression becomes: \(\frac{m^4}{1}\).

Step 3: Final Simplification

Any number divided by 1 is the number itself. Therefore, the expression \(\frac{m^4}{1}\) simplifies to \(m^4\). Since the requirement is to write answers with positive exponents, and \(m^4\) already has a positive exponent, this is our final answer.

Key Concepts

Exponent Rules

When dealing with exponents, there are fundamental rules that help simplify expressions. Understanding these rules is crucial for managing algebraic expressions effectively.

- **Zero Exponent Rule**: Whenever you have a number (other than zero) raised to the power of zero, the result is always one. For instance, if you see something like \(n^0\), you automatically know it equals 1 regardless of what \(n\) is, as long as \(n eq 0\).- **Product of Powers Rule**: If you're multiplying numbers with the same base, you simply add the exponents. For example, \(a^m \times a^n = a^{m+n}\).

- **Quotient of Powers Rule**: If you're dividing numbers with the same base, you subtract the exponent in the denominator from the exponent in the numerator: \(\frac{a^m}{a^n} = a^{m-n}\). These rules form the backbone of simplifying expressions with exponents.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and exponents, organized with operations like addition, subtraction, multiplication, and division. They are the foundation of algebra. Simplifying them often involves removing unnecessary parts and combining like terms.

A typical expression could look something like \(3x^2 + 2x - 5\). Understanding each component is key:- **Numbers (or constants)**: Numbers without attached variables, like the \(-5\) in the example.- **Variables**: Letters representing unknown values or numbers, like \(x\). - **Exponents**: Numbers indicating how many times the variable is multiplied by itself, such as \(x^2\), meaning \(x\) is multiplied by itself.By identifying and understanding each piece of an algebraic expression, you can simplify and solve equations more easily.

Positive Exponents

In algebra, expressing answers with positive exponents is often required. Positive exponents indicate regular powers and make it easier to understand the size and scale of a number.

Here are some important points:- **Understanding Positive Exponents**: A positive exponent such as \(m^4\) means you're multiplying \(m\) by itself four times: \(m \times m \times m \times m\).- **Converting Negative to Positive Exponents**: Sometimes, you might encounter negative exponents, which are reciprocals. To convert these into positive exponents, flip the base. For example, \(m^{-4}\) turns into \(\frac{1}{m^4}\).By focusing on positive exponents, you help ensure that expressions remain clear and digestible.