Problem 95 - Algebra 1

Question

Write the expression in exponential form. $$9 a \cdot 9 a \cdot 9 a \cdot 9 a \cdot 9 a \cdot 9 a$$

Step-by-Step Solution

Answer

The exponential form of the expression \(9a \cdot 9a \cdot 9a \cdot 9a \cdot 9a \cdot 9a\) is \( (9a)^6 \).

Step 1: Identify the base and the exponent

First, identify the number or the expression that is being multiplied a certain number of times. The multiplication of 9a six times suggests that \(9a\) is the base. The number of times \(9a\) is being multiplied, which is six, will serve as the exponent.

Step 2: Write the expression in exponential form

After identifying the base and the exponent, the next step is to write the expression in an exponential form structure which is base^(exponent). Therefore, the exponential form of the expression is \( (9a)^6 \).

Key Concepts

Base and Exponent

When working with exponents, it's important to understand the concepts of the base and the exponent. The base represents the number or expression that is being used repeatedly in multiplication. In our given exercise, the base is \( 9a \), which is the repeating element in the expression

• The base (\( 9a \)) is the fundamental building block of the expression.
• It's what you're multiplying by itself.

The exponent, on the other hand, tells us how many times we multiply the base by itself. Here, the expression \( 9a \) is multiplied six times. This means the exponent becomes six.

• The exponent (6) counts the occurrences of the base in the multiplication.
• It symbolizes how many times the base is used as a factor.

Putting this information together creates the exponential expression \( (9a)^6 \), compactly representing the multiplied sequence.

Multiplication

Multiplication is the arithmetic operation that combines multiple quantities together. In our example, multiplication involves combining six identical instances of the expression \( 9a \).

• Think of multiplication as adding the base to itself a certain number of times.
• In the expression \( 9a \cdot 9a \cdot 9a \cdot 9a \cdot 9a \cdot 9a \), each \( 9a \) is combined through multiplication.

Multiplying repeatedly, as seen here, is what leads to the idea of using exponents. Instead of writing out long strings of the same product, you use the base and exponent to show this repeated multiplication in a simplified form. This is particularly useful as it makes handling large numbers or extensive expressions more manageable.

Exponents and Powers

Exponents and powers are shortcuts in mathematics to express repeated multiplication. They streamline expressions by condensing them into a base raised to an exponent. In simpler terms:

• When you have an expression like \( (9a)^6 \), it's a way to say "multiply \( 9a \) by itself six times."
• Here, \( (9a)^6 \) defines a power, with 9a being the base raised to the sixth power because of its six occurrences.

Using exponents can also highlight mathematical relationships and properties between numbers, such as exponential growth and decay. Recognizing how to interpret and utilize them can add significant depth to your understanding of algebra and beyond.