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Chapter 4: Problem 2

Write each expression using exponents. $$7 \cdot 7$$

Short Answer

Expert verified
The expression \( 7 \cdot 7 \) can be written as \( 7^2 \).

Step by step solution

01

Identify Repetition

Observe the given expression: it is the product of two identical numbers, specifically 7 multiplied by 7. This indicates that the base number, 7, is repeated.
02

Rewrite Using Exponents

When the same number is repeated in multiplication, it can be expressed using exponents. Here, the number 7 is used twice as a factor, so it can be written as the base 7 with an exponent of 2: \( 7^2 \).
03

Verify the Exponential Form

Double-check the rewritten expression. The exponent of 2 denotes that the base 7 is multiplied by itself, which correctly represents the original expression of \( 7 imes 7 \).

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Key Concepts

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

Base Number
In mathematics, particularly when dealing with exponents, the term "base number" frequently arises. The base number is the number that is being multiplied by itself when using exponents. For instance, in the expression \( 7^2 \), the number 7 is the base.
  • The base number is the "foundation" of any exponential expression.
  • It tells us what number is used as a factor in repeated multiplication.
Understanding the concept of the base number is crucial as it helps us understand what value is repeatedly taken in multiplication. In our example, 7 is the base because it is repeated in the multiplication \( 7 \times 7 \). This concept lays the groundwork for simplifying expressions using exponents.
Repeated Multiplication
Repeated multiplication is a key idea behind the use of exponents. When we talk about exponents, we are essentially discussing a method to simplify the notation of multiplying the same number many times.
  • In \( 7 \times 7 \), the number 7 is multiplied by itself 2 times.
  • This repeated multiplication can be expressed as \( 7^2 \).
By converting a lengthy multiplication into an exponential form, we streamline how we write and interpret mathematical expressions. The exponent tells us how many times to multiply the base number by itself. Therefore, \( 7^2 \) is just a shorthand for saying "multiply 7 by itself once," simplifying what could be a much longer string of factors if the base were repeated many times.
Mathematical Expressions
Mathematical expressions often include exponents to convey repeated multiplication in a more manageable and concise form. By using exponents, we transform long and cumbersome multiplication sequences into understandable and simple expressions.
  • Expressions like \( 7 \times 7 \) can be neatly rewritten as \( 7^2 \).
  • The structure "base^exponent" makes complex operations simpler to interpret and solve.
These expressions are not just useful for simplification but also form the basis for many mathematical concepts, such as algebra and calculus. The power of using exponents lies in their ability to effortlessly manage and communicate large-scale repetitions and complex calculations, making it easier for both learning and application of mathematics.

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