Problem 65 - Algebra A Combined Function

Question

Simplify each expression. $$ (2 x)^{0} $$

Step-by-Step Solution

Answer

(2x)^0 simplifies to 1 using the zero exponent rule.

Step 1: Understanding the Zero Exponent Rule

Recall the zero exponent rule, which states that any non-zero number raised to the power of zero is equal to 1.

Step 2: Applying the Rule to the Expression

In the expression \((2x)^0\), "2x" is treated as a non-zero number. Therefore, applying the zero exponent rule, \((2x)^0\) simplifies to 1.

Key Concepts

Zero Exponent Rule

The zero exponent rule is a fundamental concept in algebra. It states that any non-zero number or expression raised to the power of zero equals one. This rule might seem a bit surprising at first, but it can be understood through the property of exponents which involves division of equal bases. For example, consider the expression \(a^n \div a^n\). Using the property of exponents, this simplifies to \(a^{n-n} = a^0\). On the other hand, dividing any number by itself equals one. Thus, \(a^0 = 1\). This property holds true unless the base is zero, as zero raised to any power or division by zero is undefined.
Understanding that \(2x^0 = 1\) helps simplify expressions quickly, allowing for a more efficient solving of algebraic equations.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that makes working with equations easier. It involves reducing expressions to their simplest form by combining like terms, applying exponent rules, and performing arithmetic operations. For instance, with the expression \( (2x)^0 \,\), applying the zero exponent rule reduces it directly to 1. When simplifying:

• Identify and combine like terms.
• Apply exponent rules, such as the power of a product rule and zero exponent rule.
• Simplify arithmetic operations, leaving the equation as clean as possible.

This process not only saves time but also prevents mistakes by streamlining calculations. Developing a clear understanding of simplification improves problem-solving efficiency.

Mathematical Properties

Mathematical properties, such as the zero exponent rule, provide the foundation needed to simplify and effectively work with algebraic expressions. These properties include commutative, associative, and distributive laws, each playing a crucial role in manipulating and solving expressions:

• Commutative Property: Order of addition or multiplication does not change the result.
• Associative Property: Grouping of numbers does not affect the result in addition or multiplication.
• Distributive Property: Expanding expressions like \(a(b+c) = ab + ac\)\.
• Zero Exponent Property: Any expression to the zero power equals one, as previously explained.

By understanding these properties, one can solve more complex equations with ease. Incorporating these into everyday algebraic problem-solving enhances both speed and accuracy.