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Chapter 10: Problem 56

Simplify. Write the answers with positive exponents only. $$z^{-4} z^{7}$$

Short Answer

Expert verified
The simplified expression is \(z^3\).

Step by step solution

01

Apply the Product of Powers Rule

When multiplying powers with the same base, add their exponents. Here, the base is \(z\). So, we will add the exponents of \(z\).
02

Add the Exponents

The exponents of \(z\) are \(-4\) and \(7\). Add these together: \(-4 + 7 = 3\)
03

Write the Expression with Positive Exponents

Since the exponent is positive, the simplified expression is \(z^3\).

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Key Concepts

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

Product of Powers Rule
When you multiply two powers with the same base, you need to add their exponents. This principle is known as the Product of Powers Rule. The formula for this rule is:
\(a^m \times a^n = a^{m+n}\)
Here, \(a\) is the base and \(m\) and \(n\) are the exponents.
For example, if you have \(z^{-4} z^7\), the base is \(z\). The exponents are \(-4\) and \(7\). According to the Product of Powers Rule, you add these exponents:
\(-4 + 7 = 3\)
Thus, \(z^{-4} z^7\) simplifies to \(z^3\). This rule makes it easier to solve problems involving exponents without doing the multiplication step-by-step.
Positive Exponents
An exponent tells you how many times to multiply the base by itself. A positive exponent means you keep multiplying.
For example, \(z^3\) means \(z \times z \times z\).
Positive exponents are straightforward:
  • No extra steps needed to express them.
  • They're easier to visualize and work with.
After applying the Product of Powers Rule in the exercise, we got \(z^3\). Here, the exponent is positive, so we leave it as it is.
Only negative exponents require additional steps for simplification.
Exponent Addition
Exponent addition is key when using the Product of Powers Rule. You simply add the exponents when multiplying powers with the same base.
Let's revisit the exercise:
For \(z^{-4} z^7\), you add the exponents:
\(-4 + 7 = 3\)
This step is crucial and straightforward:
  • Take the exponents.
  • Add them directly.
Adding exponents simplifies the multiplication of powers, making it easier to work with the final expression.
Always remember to keep your exponents and bases aligned to avoid mistakes.

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