Problem 46 - Algebra 1

Question

Write the expression in exponential form. nine cubed

Step-by-Step Solution

Answer

The exponential form of 'nine cubed' is \(9^3\).

Step 1: Understand What 'Cubed' Means

The term 'cubed' in mathematics means the number is multiplied by itself twice. If 'a' is a number, 'a' cubed means \(a \times a \times a\). It's the same as raising 'a' to the power of 3, which we write as \(a^3\).

Step 2: Write 'Nine Cubed' in Exponential Form

Applying the definition of 'cubed', 'nine cubed' means nine is multiplied by itself twice. In exponential form, this translates to raising nine to the power of three. We write this as \(9^3\).

Key Concepts

Cubed Numbers

In mathematics, the concept of 'cubed numbers' plays an essential role, especially when dealing with volume, geometric calculations, and algebra. Understanding what it means for a number to be cubed is fundamental.

To cube a number is to raise it to the third power, which involves multiplying the number by itself three times. For instance, to cube the number 2, you would calculate as follows: 2 cubed is equal to 2 times 2 times 2, which is equal to 8. We express this operation using the numeral followed by a small 3, also known as a superscript 3. This superscript represents the exponent.

The cubed value of a number can be understood as the volume of a cube whose sides are of equal length to the original number. For example, the volume of a cube with side lengths of 3 units is 27 cubic units, because \( 3^3 = 3 \times 3 \times 3 = 27 \). Real-world applications of cubed numbers can range from calculating the capacity of containers to determining the space inside a room.

Mathematical Notation

Mathematical notation is the system of symbols used to write down mathematical equations and expressions. It's a concise language that indicates mathematical operations, relationships, and numbers, allowing for the clear and unambiguous communication of mathematical ideas.

The use and understanding of this notation is crucial for success in mathematics. For instance, the arrangement of numbers and symbols in a particular order can dictate the procedures carried out when solving an equation or evaluating an expression. Symbols like addition (\( + \)), subtraction (\( - \)), multiplication (\( \times \)), and division (\( \div \)) are universally recognized and fundamental to mathematical notation.

Additionally, in the case of exponents, the notation further compresses expressions to denote repeated multiplication compactly. This is seen in the notation for cubed numbers — the small 3 written as a superscript is a distinct mathematical notation indicating that the number is to be multiplied by itself two additional times.

Exponents

Exponents are the power to which a number is raised, and they represent a critical element in mathematical operations. The exponent tells us how many times to multiply the base (the number) by itself.

For example, in the exponential form \( 9^3 \), the base is 9, and the exponent is 3, which signifies that you multiply 9 by itself three times: \( 9 \times 9 \times 9 \). The resulting product is 729, which is the cubed value of 9.

The use of exponents simplifies the representation and calculation of large numbers. Instead of writing out a long string of multiplication, exponents make it possible to condense this repeated multiplication into a more manageable form. Moreover, understanding how to manipulate and work with exponents is fundamental for algebra, as it allows for operations involving polynomials, exponential growth models, and other mathematical expressions to be handled more efficiently.