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Chapter 4: Problem 18

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

Short Answer

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

Step by step solution

01

Identify the repeated factor

In the expression \( k \cdot k \), the factor \( k \) is repeated twice. This means that \( k \) is multiplied by itself.
02

Write the expression using exponents

Since the factor \( k \) is repeated twice in the product, it can be written using the exponent 2. Thus, the expression \( k \cdot k \) is equivalent to \( k^2 \).

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Key Concepts

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

Understanding Repeated Factors
When you look at the expression \( k \cdot k \), you can spot that the factor \( k \) appears more than once鈥攕pecifically, it is being multiplied by itself. This is what we refer to as repeated factors. Repeatedly multiplying a number or variable is quite common in mathematics and this forms the basis of writing such expressions using exponents.

To identify repeated factors:
  • Look for elements in the expression that appear more than once.
  • Pay attention to how many times these elements (or factors) are used.
When we understand repeated factors thoroughly, we can easily transform such expressions into more manageable forms, using power notation.
Breaking Down Mathematical Expressions
A mathematical expression is a combination of symbols, numbers, and operators that show a value or relationship. In our example, \( k \cdot k \) is a simple mathematical expression that combines factors through multiplication. To better analyze these expressions:
  • Identify the operations being used (addition, subtraction, multiplication, etc.).
  • Recognize variables and constants present in the expression.
  • Observe the grouping of factors and terms.
These steps help in simplifying and restructuring expressions, making it easier to manipulate them into different forms, like converting them into exponents.
Simplifying Using Power Notation
Power notation gives us a succinct way to express repeated multiplication in mathematical expressions. In the expression \( k \cdot k \), you have a situation where the factor \( k \) is utilized repeatedly. Power notation helps streamline this repetition by using exponents:

  • Identify the repeated factor, \( k \), in the expression.
  • Count how many times \( k \) is multiplied by itself; here, it's twice.
  • Express the multiplication as an exponent: \( k^2 \).
Power notation not only simplifies the writing of such expressions but also makes it easier for us to work with complex calculations and algebraic manipulations.

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Most popular questions from this chapter

TIME Fifteen hours is what part of one day?

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{30}{37}$$

In a pen factory, an average of 5 pens out of every \(p\) pens tested are rejected. What fraction of those \(p\) pens is NOT rejected? $$\mathbf{F} \frac{5}{p}$$ $$G \frac{p}{5}$$ $$H \frac{p-5}{5}$$ $$J \frac{p-5}{p}$$

Use the Distributive Property to rewrite each expression. $$8(y+6)$$

Evaluate each expression if \(x=10, y=-5,\) and \(z=4 .\) Write as a fraction simplest form. $$\frac{1}{x^{2}}$$
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