Problem 68 - Algebra 1: Concepts and Skills

Question

WRITING POWERS Write the expression in exponential form. (Lesson \(1.2)\) \(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)

Step-by-Step Solution

Answer

The expression \(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\) in exponential form is \(3^5\).

Step 1: Identify the base

In this expression, the base is the number that is being multiplied by itself, which is 3 in this case.

Step 2: Count the number of times the base is multiplied

Count the number of times 3 is multiplied by itself. In this example, 3 is multiplied by itself 5 times.

Step 3: Write in exponential form

Write the base, which is 3, and then write the count, which is 5, as an exponent. This gives us \(3^5\).

Key Concepts

Exponents and Powers

Understanding exponents and powers is crucial for working with larger numbers, simplifying expressions, and solving various mathematical problems. In the context of mathematics, a power represents how many times a number, known as the 'base', is multiplied by itself. For instance, the expression

\(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)

shows the number 3 being multiplied five times. When you see repeated multiplication of the same number, it's much more efficient to use exponents to express this. So instead of writing the number several times, you can use a compact form – an exponent. For the exercise above, we would write \(3^5\), which reads as 'three to the fifth power' or 'three raised to the power of five'. This indicates that the base, 3, is used as a factor five times.

When using exponents, always remember that:

• The base is the number that is being repeatedly multiplied.
• The exponent signifies how many times the base appears as a factor in the multiplication.
• If there is no exponent written, it is understood to be 1, since any number to the power of 1 is itself.

Powers and exponents are not just a shorthand way to write multiplication; they also form the basis for more advanced concepts in algebra, including the laws of exponents, which allow you to manipulate expressions involving powers in useful ways.

Exponential Notation

Exponential notation is a method of writing numbers that are too large or too small to be conveniently written in standard form. Think of it as a shorthand to express repeated multiplication. Using exponential notation simplifies calculations and reduces errors in multiplication, especially with larger numbers.

In the example from the original exercise,

\(3 \cdot 3 \cdot 3 \cdot 3 \cdot 3\)

is written in exponential notation as \(3^5\). The number 3, which is the base, is the actual number we are multiplying, and the 5, which is the exponent, tells us the number of times to multiply the base by itself. This notation is not only simpler but essential for working with powers in algebra, science, and technology.

Exponential notation is also commonly used in scientific notation to represent very large or very small numbers, making it easier to handle the vast range of values that can occur in scientific calculations.

Base and Exponent

At the heart of exponential notation are the concepts of 'base' and 'exponent'. The base is the number that is being multiplied, and the exponent tells us how many times to multiply the base by itself.

For example, in the expression \(3^5\),

the base is 3, and the exponent is 5. The base can be any real number, and the exponent indicates the number of times the base appears as a factor. It's also important to understand that exponents can be more than just positive integers. They can be zero, negative numbers, or even fractions, which expands the utility of exponentiation when dealing with various mathematical concepts such as roots and growth/decay rates.

To put it simply:

• The base is what we start with.
• The exponent tells us how many times to use the base in the multiplication.

It's this simple yet powerful concept that allows us to write and compute with large numbers more efficiently and to understand exponential growth and decay in real-world contexts.