Problem 12 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ -6^{0} $$

Step-by-Step Solution

Answer

The expression \(-6^{0}\) evaluates to \(-1\).

Step 1: Understanding Exponents

Exponents denote how many times a number, known as the base, is multiplied by itself. Here, the base is 6, and the exponent is 0. According to the laws of exponents, any non-zero number raised to the power of 0 is 1.

Step 2: Apply Exponent Rules

Use the exponent rule: for any non-zero number, if raised to the power of zero, the result is 1. Therefore, we have 6^0 = 1.

Step 3: Consider the Negative Sign

The problem states -6^0. According to the order of operations, exponents are evaluated before negation. Therefore, first, compute 6^0 = 1. Now apply the negativeness: -1.

Step 4: Final Evaluation

Combine all calculations made above. Therefore, -6^0 is evaluated as -1

Key Concepts

Exponents

An exponent is a small number placed to the upper right of a base number. It indicates the number of times the base is multiplied by itself. For example, in the expression \( a^b \), \( a \) is the base and \( b \) is the exponent. This is read as "\( a \) raised to the power of \( b \)." If \( b = 3 \), it simply means \( a \times a \times a \).
It's important to know certain laws of exponents to handle these calculations efficiently:

• Any number raised to the power of zero is 1, that is \( a^0 = 1 \) for any \( a eq 0 \).
• To multiply exponents with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Division of exponents with the same base means you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In our example, \( 6^0 \) is given, it follows the rule that any number to the power of zero equals 1, regardless of what the base is, as long as it is not zero.

Base and Exponent

In any expression involving exponents, understanding the roles of the base and exponent is crucial. The base is the main number that is multiplied by itself, while the exponent shows how many times this multiplication occurs.
Take \( 6^0 \) for instance. Here, \( 6 \) is the base, and \( 0 \) is the exponent. Despite needing multiplication, any number raised to the power of 0 always equals 1, as per exponent rules.
The concept of base and exponent is an integral part of mathematical calculations because:

• The base determines the fundamental number you start with.
• The exponent tells you how many times the base is used in the multiplication process.

Grasping this concept helps in breaking down complex calculations into manageable steps. For example, instead of multiplying \( 6 \, \times 6 \, \times 6 \), you can write and calculate \( 6^3 \).

Negative Numbers

Negative numbers are values less than zero, and they are crucial in mathematics, including when using the order of operations. They are often paired with positive numbers and are shown with a minus sign. In our exercise involving \(-6^0\), it's critical to emphasize how negative numbers interact with exponentiation and order of operations.
The expression \(-6^0\) should be read as \(-1 \times 6^0\). According to the order of operations, exponents are handled before multiplication or negation. Therefore, you first calculate \(6^0\), which is 1, and then apply the negative sign to obtain the result of \(-1\).
In summary:

• Calculate the exponent value first.
• Apply the negative sign to the result of the exponentiation.

Understanding negative numbers in context with exponents allows for clearer and precise calculations, especially when dealing with order of operations and complex expressions.