Problem 26 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x^{7} y^{0}$$

Step-by-Step Solution

Answer

The simplification of \(x^{7} y^{0}\) is \(x^{7}\).

Step 1: Write down the expression

Let's write down the expression, it is \(x^{7} y^{0}\).

Step 2: Apply the zero-exponent rule

Know that any number except zero raised to the power zero is one. So \(y^{0}\) equals to 1.

Step 3: Simplify the Expression

Replace \(y^{0}\) with 1 in the expression. Thus, the expression \(x^{7} y^{0}\) simplifies to \(x^{7} * 1\), which is \(x^{7}\).

Key Concepts

Zero-Exponent Rule

One important property of exponents is the zero-exponent rule. This rule states that any non-zero number raised to the power of zero is equal to one.
This can be a bit unintuitive at first, but it simplifies many expressions and helps in calculations immensely.
Why does this work? Imagine dividing a number by itself, say \( a^3 \) divided by \( a^3 \). You get \( a^{3-3} = a^0 = 1 \). So, it simplifies to one as long as the base is not zero.
Be sure to apply this rule correctly to avoid mistakes:

• Applies to non-zero bases.
• Helps in reducing complex expressions.
• Avoids needless terms in equations, making algebra easier.

Remembering this rule will help you work through exponential expressions efficiently.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. When dealing with exponents, this often means using exponent rules to eliminate terms that can be easily resolved, like the zero exponent term.
For example, in the expression \( x^7 y^0 \), you simplify by applying the zero-exponent rule.
This changes \( y^0 \) to 1, letting you focus on \( x^7 \) only. Breaking down complex expressions to simpler parts makes them easier to handle and understand:

• A simplified expression is usually easier to evaluate.
• Lowers the risk of errors in calculations.
• Gives a clearer view of the problem you are solving.

Always check your work for possible areas to simplify, especially when dealing with multiple terms.

Exponents

Exponents are the small numbers placed above and to the right of a base number, indicating how many times the base is multiplied by itself.
In our example, \( x^7 \) shows that \( x \) is multiplied by itself seven times.
Exponents provide a way to write repeated multiplication concisely and are critical in higher mathematics:

• Aids in simplifying complex multiplication tasks.
• Works with other algebraic rules to solve equations.
• Allows expression of very large or very small numbers easily, using scientific notation.

Understanding the basic properties and rules of exponents is key for working with algebraic expressions.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols.
It is used to express mathematical relationships and solve equations involving unknown values.
Algebra uses variables like \( x \) and \( y \) to represent numbers in a general form, which enables more flexible problem-solving.
In the expression \( x^7 y^0 \), algebraic techniques along with exponent rules help simplify the expression effectively:

• Variables allow for general solutions, applicable across various scenarios.
• Combining rules, like exponent rules, simplifies solving complex problems easier.
• Helps in modeling real-world situations with equations.

Learning algebra provides a foundation for advanced study in other fields of mathematics and sciences.