Problem 71 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 10^{0} $$

Step-by-Step Solution

Answer

The evaluated expression \(10^{0}\) is 1.

Step 1: Applying the zero exponent rule.

In mathematics, the zero exponent rule states that any non-zero number raised to the power of 0 equals 1. Thus, in our case, \(10^{0} = 1\).

Step 2: Identify the operation required

Determine what computation or simplification is needed.

Step 3: Perform the calculation

Execute the computation step by step.

Step 4: Simplify and state the result

Simplify and write the final answer.

Step 5: Conclude with the answer

The evaluated expression \(10^{0}\) is 1.

Key Concepts

Exponent Rules

Understanding how exponents work is fundamental in math, particularly in algebra. Exponents dictate how many times a number, known as the base, is multiplied by itself. For example, the expression \(5^3\) means that you multiply 5 by itself 3 times: \(5 \times 5 \times 5\) which equals 125.

There are several key rules when it comes to working with exponents:

• Product of Powers: When multiplying two exponents with the same base, you add the exponents together (\(a^m \times a^n = a^{m+n}\)).
• Quotient of Powers: When dividing two exponents with the same base, you subtract the exponent in the denominator from the exponent in the numerator (\(a^m \/ a^n = a^{m-n}\)).
• Power of a Power: When raising an exponent to another exponent, you multiply the exponents (\( (a^m)^n = a^{m \times n}\)).
• Power of a Product: When raising a product to an exponent, the exponent applies to each factor within the parentheses (\( (ab)^n = a^n \times b^n \)).
• Zero Exponent: Any non-zero base raised to the power of zero equals 1 (\(a^0 = 1\)) regardless of the value of \(a\).
• Negative Exponent: A negative exponent indicates a reciprocal; \(a^{-n}\) is equivalent to \(1/a^n\).

A solid grasp of these rules simplifies the process of evaluating expressions with exponents and allows for the accurate solving of algebraic equations.

Evaluating Expressions

When we talk about evaluating mathematical expressions, we're referring to the process of calculating the value of the expression, often based on certain rules. Evaluating can include simplifying expressions, solving for unknown values, or substituting values into variables. The key is to follow the correct order of operations, which in most cases is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS.

For example, evaluating the expression \(2 + 4^2 \times 3\) means you would first handle the exponent (raising 4 to the power of 2), then the multiplication (multiplying by 3), and finally, perform the addition (adding 2). The sequence matters because if you do the steps out of order, you can end up with a completely different result.

Mathematical Properties

The principles that underpin the operations and expressions in mathematics are called mathematical properties. These properties are the backbone of all algebraic manipulations and solutions. Some of the basic properties include:

• Commutative Property: This property states that you can change the order of the numbers when adding or multiplying without changing the result (\(a + b = b + a\) and \(a \times b = b \times a\)).
• Associative Property: When adding or multiplying, it does not matter how we group the numbers (\( (a + b) + c = a + (b + c)\) and \( (a \times b) \times c = a \times (b \times c)\)).
• Distributive Property: Multiplication distributes over addition (\(a \times (b + c) = (a \times b) + (a \times c)\)).
• Identity Property: Adding 0 to a number doesn't change its value (\(a + 0 = a\)), and multiplying a number by 1 also does not change its value (\(a \times 1 = a\)).
• Inverse Property: Adding the additive inverse (the opposite number) equals zero (\(a + (-a) = 0\)), and multiplying by the reciprocal equals one (\(a \times 1/a = 1\), for \(a \eq 0\)).

These properties are universally accepted and apply to all real numbers, facilitating the ability to evaluate expressions, simplify calculations, and prove various mathematical theorems and identities.