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Chapter 1: Problem 20

Write each product using exponents. See Example \(1 .\) $$ b \cdot b \cdot b \cdot b $$

Short Answer

Expert verified
The product is written as \( b^4 \).

Step by step solution

01

Identify the Base

The base of an expression with exponents is the number or variable that is repeatedly multiplied. Here, the base is the variable \( b \) since it is repeated in the multiplication.
02

Count the Frequency

Count how many times the base \( b \) is multiplied by itself. In the expression \( b \cdot b \cdot b \cdot b \), the base \( b \) is used 4 times.
03

Write with Exponents

Combine the base from Step 1 and the frequency from Step 2 to write the expression using exponents. We express \( b \cdot b \cdot b \cdot b \) as \( b^4 \).

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Key Concepts

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

Base and Exponent Identification
Understanding base and exponent identification is crucial when working with expressions in mathematics. In expressions involving exponents, the **base** is the number or variable that is multiplied by itself a certain number of times.
Consider the expression \( b \cdot b \cdot b \cdot b \). In this case:
  • The base is \( b \), since it is the element being repeated in the multiplication.
Once the base is identified, the next step is to recognize the **exponent**. The exponent signifies how many times the base is multiplied by itself.
  • In \( b \cdot b \cdot b \cdot b \), the base \( b \) appears 4 times.
  • Therefore, the exponent is 4.
This method simplifies the expression to \( b^4 \), where \( b \) is the base and 4 is the exponent, highlighting the repetition in a compact form.
Multiplying Variables
An essential part of working with algebraic expressions is knowing how to multiply variables efficiently. When you encounter a series of variables being multiplied, look for repetition.
Consider the expression \( b \cdot b \cdot b \cdot b \):
  • All the factors are the same, being the variable \( b \).
To simplify repeated multiplication, use exponents. This involves two steps:
  • Identify the repeating variable (the base).
  • Count how many times it's repeated (for the exponent).
Let's say you are given \( b \cdot b \), you know:
  • It can be rewritten as \( b^2 \) since \( b \) is multiplied by itself twice.
This rule helps in simplifying even large expressions, making them easier to handle and solve.
Writing Expressions with Exponents
Writing expressions with exponents not only makes equations more readable but also allows for easier manipulation in calculations. Exponents encapsulate the idea of repetitive multiplication within a concise format.
Taking an expression like \( b \cdot b \cdot b \cdot b \), the goals are:
  • To simplify the representation using the smallest number of characters.
  • To capture the repetition in a single notation: \( b^4 \).
Here's how to write expressions using exponents effectively:
  • Identify the repeated base (here it is \( b \)).
  • Determine the exponent based on the frequency of repetition (in this case, 4).
By rewriting multiplication as an exponential expression, you manage complexity and make subsequent mathematical operations more manageable.

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Most popular questions from this chapter

Recall that the perimeter of a figure is equal to the sum of the lengths of its sides. First Aid. Each side of the red cross has length \(x\) inches. Write an algebraic expression that represents the perimeter of the cross.

Simplify each expression, if possible. $$ -v-3 v+6 v+2 v $$

Simplify by combining like terms. $$ 10 y^{2}-8 y+y-7 $$

Explain each error. Then give the correct answer. a. \(9(4 b-2)=36 b-2\) b. \(\quad 3(2 x)=6 \cdot 3 x=18 x\) c. \(-(23 c+2)=-23 c+2\) d. \((5 n+1) 2=5 n+2\)

Simplify each expression, if possible. $$ -(c+7)+2(c-3) $$
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