Problem 23 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(5^{3}\right)^{-1}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{125}\).

Step 1: Evaluate the Exponent

First, evaluate the exponent inside the brackets. The expression is \(5^3\), which means \(5\) is multiplied by itself two more times: \(5 \times 5 \times 5 = 125\).

Step 2: Apply Negative Exponent Rule

The given expression is \( (5^3)^{-1} \). According to the negative exponent rule, \(a^{-n} = \frac{1}{a^n}\). Applying this to the expression gives us \(\left(5^3\right)^{-1} = \frac{1}{5^3}\).

Step 3: Substitute the Evaluated Exponent

Now, substitute the evaluated result from Step 1 into the expression from Step 2: \(\frac{1}{5^3} = \frac{1}{125}\). Hence, the expression simplifies to \(\frac{1}{125}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number being multiplied, while the exponent tells you how many times to multiply the base by itself. It's often written in the form of "base" raised to the "exponent", such as \(5^3\). This means multiplying five by itself, two additional times: \(5 \times 5 \times 5 = 125\). Exponents are incredibly useful as they allow us to express large numbers in a compact form. It's important to understand both positive and negative exponents, as they have distinct roles in mathematical expressions. A positive exponent signifies repeated multiplication, making the number larger, while a negative exponent indicates division, effectively reducing the number's size.
Negative exponents can sometimes be confusing. Generally, when you have a negative exponent, it signifies the reciprocal of the base raised to the opposing positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\). In simple terms, a number with a negative exponent is like flipping it upside down in fraction terms.

Simplification of Expressions

Simplification of expressions involves reducing a mathematical expression to its simplest form, making it easier to understand or further manipulate. This process often includes evaluating exponents, applying mathematical rules, and simplifying fractions. When you simplify an expression, you want to break it down into fewer terms with simpler numbers.
To effectively simplify expressions involving exponents, you must be familiar with specific rules like the power of a product, power of a power, and the negative exponent rule. Applying these principles helps in methodically reducing expressions. In our exercise, we initially have \((5^3)^{-1}\).

• Evaluate the exponent (\(5^3 = 125\)).
• Apply the negative exponent rule (\(\frac{1}{5^3}\)).
• Substitute the evaluation (\(\frac{1}{125}\)).

Following these steps methodically is key to simplifying expressions cleanly and ensuring accuracy.

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators (like \(+, -, \times, \div\)) designed to express a value or relationship. They are the language of math, allowing us to describe and analyze numerical relationships concisely. Working with these expressions can often involve several operations including addition, subtraction, multiplication, division, and exponentiation.
Understanding the components of mathematical expressions is critical. Each element, whether a number or an operator, plays a role in creating the structure and determining the order of operations. In simplifying expressions, closely following the order of operations—parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right), often remembered by "PEMDAS"—is crucial.
Our original expression, \((5^3)^{-1}\), is an example of using these rules to reach a simplified form, \(\frac{1}{125}\). By understanding each expression component, you can execute calculations properly and enhance your mathematical accuracy, making your work much more efficient and comprehensive.