Problem 72 - Algebra 1: Concepts and Skills

Question

WRITING POWERS Write the expression in exponential form. (Lesson \(1.2)\) twelve cubed

Step-by-Step Solution

Answer

The given phrase 'twelve cubed' can be written in exponential form as \(12^3\).

Step 1: Understand the Words in the Problem

The word 'cubed' in mathematics usually refers to raising a number to the power of three. It is one of the few powers that has its own term. Similarly, 'squared' means to the power of two.

Step 2: Translate into Mathematical Notation

To write 'twelve cubed' in mathematical notation, you write the number 12 (the base) and then write the number 3 (the exponent) as a superscript. It is read as 'twelve to the power of three'.

Step 3: Write the Final Answer

The final answer in exponential form will be \(12^3\).

Key Concepts

Exponential Form

Exponential form is a way of writing numbers that involves using a base and an exponent. This is a compact and efficient method to represent repeated multiplication of the same number. For example, if you want to express "twelve cubed", you write it in exponential form as \(12^3\). This means multiplying the base number, 12, by itself a total of three times:

• 12 is the base number, which is repeated in multiplication.
• The exponent, 3, indicates how many times the base is used as a factor.

Understanding exponential form helps simplify expressions and makes it easier to work with much larger numbers. It also provides a clear and concise way to express very small or very large values in mathematics. This form is commonly used in algebra, science, engineering, and everyday calculations.

Powers

The term "power" in mathematics refers to the expression of a number that has been multiplied by itself a certain number of times. When you hear terms like "squared" or "cubed", they indicate specific powers.

• When a number is "squared", it means it is raised to the power of two, like \(a^2\).
• "Cubed" means it is raised to the power of three, so \(12^3\) is read as twelve cubed.

Powers are a vital part of understanding exponential expressions because they inform us how many times we multiply the base by itself. Powers make calculations easier, especially when dealing with large exponents. Furthermore, recognizing different powers by their names, like squared and cubed, can be handy in both problem solving and in the real-world where these terms are commonly used.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to write down mathematical ideas and quantities clearly and concisely. It provides a universal language for mathematicians to communicate complex concepts efficiently. In the context of exponents, mathematical notation helps to systematically express how numbers are multiplied together. For example, in the expression \(12^3\):

• The superscript number 3 is the exponent indicating that the number 12 is multiplied by itself three times.
• The base number, 12, is written normally and represents the number being multiplied.

This notation is not just limited to whole numbers; it can be applied to fractions, variables, and negative numbers as well. Understanding mathematical notation is crucial for solving a wide range of problems in algebra and beyond. It allows students to interpret mathematical expressions accurately and consistently.