Problem 47 - Elementary Algebra

Question

\((5 x)^{0}\)

Step-by-Step Solution

Answer

1

Step 1: Understand the Zero Exponent Rule

Any non-zero number raised to the power of 0 is 1. This is known as the zero exponent rule. In mathematical terms, if \(a\) is any non-zero number, then \(a^{0} = 1\).

Step 2: Apply the Zero Exponent Rule

In the given expression \((5x)^{0}\), recognize that it follows the zero exponent rule because \(5x\) is a non-zero term. Therefore, \((5x)^{0} = 1\).

Key Concepts

Zero Exponent Rule

The Zero Exponent Rule is a fundamental concept in algebra and exponents. This rule states that any non-zero number raised to the power of 0 is equal to 1.
In mathematical notation: \( a^{0} = 1 \) where \( a \) is any non-zero number.
This rule helps simplify expressions and solve equations by reducing complexities. Let's demonstrate it with an example:
Given the expression \( (5x)^{0} \), applying the zero exponent rule, we have: \( (5x)^{0} = 1 \). This is because \( 5x \) is a non-zero term.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number by itself.
For example, \( 2^3 \) means \( 2 \) multiplied by itself three times: \[2 \times 2 \times 2 = 8 \].

Exponents have various rules that make calculations easier:

• Product of Powers: \ a^m \times a^n = a^{m+n} \
• Power of a Power: \ (a^m)^n = a^{m \times n} \
• Quotient of Powers: \ \frac{a^m}{a^n} = a^{m-n} \

Understanding these rules can help simplify algebraic expressions and solve exponential equations more efficiently.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is essential for solving equations and understanding mathematical relationships.

Key concepts in algebra include:

• Variables: Symbols (usually letters) that represent numbers or values
• Expressions: Combinations of variables and constants using operations (e.g., \( 3x + 5 \))
• Equations: Mathematical statements that assert the equality of two expressions (e.g., \( 2x + 3 = 7 \))

Algebraic rules, such as the zero exponent rule, are crucial for simplifying and solving these expressions and equations. By mastering algebra, students can tackle more complex mathematical problems with confidence.