Question

Without a calculator, decide whether the quantities are positive or negative. $$ -48^{0} $$

Short answer

Answer: The given quantity $$-48^0$$ is positive.

Step-by-step solution

Recall the exponent rule for any non-zero number raised to the power of 0

According to the exponent rules, any non-zero number raised to the power of 0 is equal to 1. In other words, $$a^0 = 1$$ for any non-zero number $$a$$.

Apply the exponent rule to the given quantity

In our case, the given quantity is $$-48^0$$. Since -48 is a non-zero number, we can apply the exponent rule mentioned in step 1: $$-48^0 = 1$$

Determine the sign of the result

Now that we have found the value of the given quantity, we can determine its sign. Since the value is 1, which is a positive number, the given quantity is positive.

Key concepts

Exponent Rules

Exponent rules are essential for simplifying mathematical expressions involving repeated multiplication. One critical rule to comprehend is the zero exponent rule. According to this rule, any non-zero number raised to the power of zero equals one. This applies because the idea behind exponents is to multiply a number by itself a certain number of times. When the exponent is zero, it suggests multiplying the number zero times, which results in one.
To put it simply, for any non-zero number \( a \), the equation \( a^0 = 1 \) holds true. This rule is consistent and widely applicable, whether dealing with large numbers or algebraic expressions. Understanding this concept means the ability to simplify expressions efficiently and correctly, contributing to solving more complex math problems effectively.

Positive and Negative Numbers

When dealing with expressions, understanding the signs—positive and negative—of numbers is crucial. A positive number is greater than zero, while a negative number is less than zero. The transition between different signs can significantly alter the value of a mathematical expression.
In the context of the given problem, the expression \(-48^0\) involves a zero exponent applied to a negative number.

• The expression \(-48^0\) represents a negative number "-48" raised to the zero power.
• Applying the zero exponent rule, any non-zero number powered by zero equals 1. So although originally negative, the outcome here is positive 1.

Moreover, understanding signs is foundational in algebraic expressions, impacting everything from basic arithmetic visualizations to complex algebraic problem-solving.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Variables serve as symbols representing values that can change. The expressions may involve addition, subtraction, multiplication, division, and powers (exponents).
Grasping exponentiation within algebraic expressions is crucial, as it dictates how terms combine and simplify. The zero exponent plays a pivotal role, allowing simplifications that can reduce complexity. For example, in simplifying \(-48^0\), the expression itself simplifies based on the zero exponent principle to a decisive result of 1.Handling such expressions requires familiarity with operator precedence, which dictates the sequence of execution. Parentheses are also vital in maintaining clarity and ensuring accurate application of operations. This knowledge empowers students to navigate algebraic problems with confidence, ensuring accurate results and fostering a deeper grasp of mathematical concepts.