Question

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

Short answer

Answer: The value of \(0^3\) is 0.

Step-by-step solution

Understand the exponent

We are given the expression \(0^3\), which means 0 raised to the power of 3. In general, any number (except 0) raised to the power of 0 is equal to 1. However, when 0 is raised to any power greater than 0, the result is always 0.

Evaluate the expression

So, in this case, since 0 is raised to the power of 3, we have: $$ 0^3 = 0 \times 0 \times 0 $$

Simplify

Multiply the numbers together and simplify the expression: $$ 0 \times 0 \times 0 = 0 $$ Thus, the value of the expression \(0^3\) is 0.

Key concepts

Zero Exponent Rule

The zero exponent rule is a crucial concept in understanding exponents. It states that any non-zero number raised to the power of zero equals one. This can often surprise students, but there's a logical explanation behind it. For instance, consider repeatedly dividing a number by itself:

• Start with a number, say, \(a^3 = a \times a \times a\).
• Dividing by the number each step lowers the exponent. Hence, \(a^3 \div a = a^2\).
• Continuing, \(a^2 \div a = a\).
• Finally, \(a \div a = 1\), so \(a^0 = 1\).

This pattern shows why any number to the zero power is 1. But remember, zero to the power of zero is a special case and typically left undefined.

Power of a Number

Raising a number to a power means multiplying it by itself a specific number of times. This is commonly referred to as an "exponent." The exponent tells us how many times the number (the base) has to be multiplied by itself.
For example, in \(5^3\), the number 5 is the base, and 3 is the exponent. This expression indicates that 5 must be multiplied by itself two more times:

• \(5 \times 5 \times 5 = 125\).

This process drastically increases the value of numbers, especially when using larger bases or higher exponents. Understanding how powers work is foundational for algebra and beyond, as it's necessary for solving exponentially growing problems.

Multiplication

Multiplication is a basic arithmetic operation that represents the concept of adding a number to itself a certain number of times. When dealing with exponents, multiplication is often used to evaluate expressions. Let's consider \(0^3\) as an example:

• The base 0 needs to be multiplied by itself three times.
• This can be shown as \(0 \times 0 \times 0\).
• Multiplying any number with 0 results in 0, which simplifies the expression to simply 0.

This principle applies universally with zero. When multiplying zero by any number, the result is always zero. Understanding this can significantly simplify problems in mathematics involving zero, especially in contexts of powers and multiplication.