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Chapter 0: Problem 19

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

Short Answer

Expert verified
The expression is \( z^7 \).

Step by step solution

01

Identify the base

Notice that the base being repeatedly multiplied is the same number, which is 'z'.
02

Count the number of factors

Count how many times 'z' is being multiplied by itself. In this expression, 'z' appears 7 times.
03

Write the expression using exponents

Using the base 'z' and the number of times the base is multiplied (7), we can write the expression as an exponent. This gives us the exponent form: \[ z \times z \times z \times z \times z \times z \times z = z^7 \]

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Key Concepts

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

base in exponents
When dealing with exponents in algebra, one key idea is understanding the concept of the 'base.' The base in an exponent is the number or variable that is being repeatedly multiplied.

In the expression given:
\[ z \times z \times z \times z \times z \times z \times z \]
The base is 'z', because 'z' is the value that is being multiplied multiple times.
Having a clear grasp on what the base is, makes it easier to transition into writing and simplifying expressions using exponents.
multiplication of same base
When you multiply the same base in algebra, you combine the repeated factors into a single term using an exponent.

For instance:
In the expression:
\[ z \times z \times z \times z \times z \times z \times z \]
Each 'z' is being multiplied by itself, which is why you can utilize exponents to simplify this expression.
In this case, you count how many times the base 'z' is multiplied. Here, it is multiplied 7 times.
To represent this multiplication of the same base in a simpler form, you write:
\[ z^7 \]
This demonstrates that the base 'z' is used 7 times in the multiplication.
writing expressions with exponents
Learning how to write expressions with exponents is an important skill in algebra. It allows you to simplify long multiplication sequences.

Consider the expression:
\[ z \times z \times z \times z \times z \times z \times z \]
This lengthy multiplication can be compactly written with an exponent. Here's how you can do it:
  • First, identify the base. In this expression, the base is 'z'.
  • Next, count the number of times the base is multiplied. In this case, it is 7 times.
  • Finally, use the exponent to express this multiplication. You write the base followed by the exponent, which tells you how many times to multiply the base by itself.

Thus, the expression becomes:
\[ z^7 \]
This method of writing expressions using exponents makes it easy to manage and understand large quantities of repeated multiplication.

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Most popular questions from this chapter

Simplify each expression. $$ 7-\frac{3}{4}(-8)+12 \cdot \frac{5}{6} $$

Evaluate each expression for \(a=-3, b=64,\) and \(c=6 .\) $$ 3 a+\sqrt{b} $$

Write each fraction as a decimal. For repeating decimals, write the answer by first using bar notation and then rounding to the nearest thousandth. See Example 11. $$ \frac{8}{9} $$

Use an inequality symbol to write each statement. $$ 5+0 \text { is not equal to } 0 $$

Simplify each expression. $$ 6|4-5|-24 \div 3 $$
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