Question

Evaluate the expressions without using a calculator. $$ 3^{0} $$

Short answer

Answer: The value of the expression 3^0 is 1.

Step-by-step solution

Evaluate the expression 3^0

According to the basic exponent rule, any non-zero number raised to the power of 0 is equal to 1. $$ 3^{0} = 1 $$ So, the result of the given expression is 1.

Key concepts

Zero Exponent

Exponentiation is a mathematical operation involving two numbers, a base and an exponent. When the exponent is zero, it is known as the "zero exponent" rule. This may seem confusing at first, but it's quite simple:- Any non-zero base raised to the power of zero equals one.- This rule holds true regardless of whether the base is positive or negative.To better understand why this happens, consider the pattern in decreasing exponents:- When you divide a base by itself, like in \(3^1 \,/\, 3^1\), you're left with \(3^0\).- This division equals \(1\) since \(3^1\,=\,3\) divided by \(3\,=\,3\) results in \(1\).So, \(3^0 \ = \ 1\), and that's the magic of the zero exponent! This rule is incredibly useful in simplifying expressions involving different exponents.

Evaluate Expressions

Evaluating expressions is an essential skill in algebra and higher mathematics. It means finding the value of an expression by performing the required operations. Without a calculator, this involves understanding the rule that applies to the given operation.In our original exercise, the expression \(3^0\) is evaluated using the zero exponent rule:- The base number is \(3\).- The exponent is \(0\).Using the zero exponent rule—since any base raised to the zero power is equal to one—you can quickly determine that \(3^0\ = \ 1\). It's important to recognize these rules to simplify and evaluate expressions efficiently.

Basic Exponent Rule

Exponent rules are foundational in simplifying mathematical expressions. These rules allow us to handle expressions with powers efficiently, without lengthy calculations.The basic exponent rule we are focusing on is the zero exponent rule, which states:- \((a^0 = 1)\) for any non-zero number \(a\).This rule becomes a powerful tool because:- It saves time by making complicated expressions simple.- It provides a consistent method for evaluation amongst various expressions.For example, \(2^0, \, -5^0, \ (1/2)^0\), all simplify to \(1\) effortlessly. Knowing these rules can significantly make problem-solving easier, especially in algebra and calculus.