Question

Rewrite the expression with positive exponents and simplify.\((x+5)^{0}, \quad x \neq-5\)

Short answer

The simplified expression is 1.

Step-by-step solution

Apply the Zero Exponent Rule

The Zero Exponent Rule states that any nonzero number raised to the power of 0 is 1. Therefore, apply this rule to simplify the expression \((x+5)^0\). Regardless of the value of \(x\), the whole expression equals 1.

Key concepts

Positive Exponents

Understanding positive exponents is crucial for simplifying mathematical expressions. Positive exponents indicate the number of times a base is multiplied by itself. For example, the expression \(2^3\) represents \(2 \times 2 \times 2\), which equals 8. In contrast to negative exponents, which imply division, positive exponents reflect multiplication. It’s vital to remember that any base, except zero, raised to the power of zero is always one, which leads us to the zero exponent rule—a special case in the realm of exponents.

When you encounter expressions with positive exponents, simplify them by performing the multiplication denoted by the exponent. For example, simplifying \(x^2\) when \(x=3\) involves calculating \(3^2 = 3 \times 3 = 9\). Positive exponents are straightforward and play a fundamental role in understanding more complex exponential rules.

Simplifying Expressions

Role of Simplification

Simplifying expressions is a fundamental process in algebra that makes equations easier to work with. Simplification can involve combining like terms, reducing fractions, or applying exponent rules. By simplifying an expression, we aim to transform it into its most concise and straightforward form without changing its value.

Simplifying with Exponent Rules

Specifically, when simplifying expressions with exponents, it is crucial to employ exponent properties correctly. If you come across \( (x+5)^0 \) and we know from the zero exponent rule that anything to the power of zero is one, the simplification process leads us to conclude that the expression equals 1, given that the base \(x+5\) is not zero. The simplification step transforms potentially complicated expressions into something far more manageable and often reveals a clearer understanding of the relationship between the variables involved.

Exponent Properties

Exponents follow several key properties that make it possible to manipulate and simplify expressions involving powers. Among these properties are:

• The Product Rule, which states that when multiplying like bases, you can add the exponents (e.g., \(a^m \times a^n = a^{m+n}\)).
• The Quotient Rule, informing us that when dividing like bases, we subtract the exponents (e.g., \(a^m \div a^n = a^{m-n}\)).
• The Power of a Power Rule, which says that to raise a power to another power, you multiply the exponents (e.g., \( (a^m)^n = a^{mn}\)).
• The Zero Exponent Rule we encountered in the original exercise, which asserts that any nonzero base raised to the zero power is equal to one.

These properties allow us to tackle complex expressions and reduce them to more simple and solvable forms. It's important to note that these rules help ensure consistency across operations and provide a structured approach to solving problems that involve exponents. When applied correctly, exponent properties are powerful tools in algebra that enable students to unlock solutions efficiently and effectively.