Problem 75 Write using exponential notation... [FREE SOLUTION]

Chapter 10: Problem 75

Write using exponential notation. $$ 2 \cdot 2 \cdot a \cdot a \cdot a \cdot a $$

Short Answer

Expert verified

The expression in exponential notation is $2^2 \cdot a^4$.

Step by step solution

Identify Repeated Factors

First, we examine the given expression: $2 \cdot 2 \cdot a \cdot a \cdot a \cdot a$. Notice that the number $2$ is repeated twice and the variable $a$ is repeated four times.

Express Using Exponents

Next, we rewrite the repeated factors using exponential notation. Since $2$ appears twice, we write it as $2^2$. The variable $a$, which appears four times, is written as $a^4$.

Combine the Terms

Finally, we combine these expressions to write the entire term in exponential notation: $2^2 \cdot a^4$. This concisely represents the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Repeated Factors

When working with mathematical expressions, repeated factors are numbers or variables that appear more than once in a multiplication sequence. Understanding how to identify these repeated factors is crucial for simplifying expressions. Let's break it down:

Consider the sequence of numbers or variables, like in the expression given: $2 \cdot 2 \cdot a \cdot a \cdot a \cdot a$.
The goal is to recognize which elements are repeated. Here, the number $2$ occurs twice, and the variable $a$ is repeated four times.
Identifying these repetitions is the first step to simplifying the expression using exponential notation, a more concise form.

By spotting repeated factors, you lay the groundwork for expressing them with ease using exponents.

Exponents

Exponents are a powerful mathematical tool that allows us to write repeated factors in a compact form. They reflect the number of times a number or variable is multiplied by itself. Once repeated factors are identified, using exponents is the next step:

An exponent is written as a small number (called the power) above and to the right of the base number or variable.
In our expression, the number $2$ is written as $2^2$ because it appears two times: $2 \cdot 2 = 2^2$.
Similarly, the variable $a$ is represented as $a^4$ since it repeats four times: $a \cdot a \cdot a \cdot a = a^4$.

Therefore, exponents let us write long multiplications in a shorter, more manageable form. This simplification is particularly helpful in complex expressions and equations.

Prealgebra

Prealgebra sets the foundation for all future math courses, making it crucial for students to understand core concepts like repeated factors and exponents. In prealgebra, students learn:

Basic operations including multiplication, which is often where repeated factors arise.
How to simplify expressions, like reducing $2 \cdot 2 \cdot a \cdot a \cdot a \cdot a$ to $2^2 \cdot a^4$ using exponentiation.
The significance of rewriting expressions in simpler forms, which is essential for solving equations and inequalities.

This subject acts as a bridge, connecting fundamental arithmetic skills to more advanced algebraic concepts. A solid grasp of these basics not only aids in understanding math today but also paves the way for success in algebra and beyond.

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91影视

Write using exponential notation. $$ 2 \cdot 2 \cdot a \cdot a \cdot a \cdot a $$

Short Answer

Step by step solution

Identify Repeated Factors

Express Using Exponents

Combine the Terms

Key Concepts

Repeated Factors

Exponents

Prealgebra

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