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Chapter 5: Problem 19

Simplify each numerical expression. \(10^{-1} \cdot 10^{-2}\)

Short Answer

Expert verified
The simplified expression is \(\frac{1}{1000}\).

Step by step solution

01

Recall the Rule of Exponents

To simplify expressions involving exponents, we can use the rule that states when multiplying like bases, you add the exponents. This rule is given by: if you have \( a^m \times a^n \), this simplifies to \( a^{m+n} \). Here, the base is 10.
02

Apply the Rule to the Given Expression

Apply the rule from Step 1 to the expression \(10^{-1} \cdot 10^{-2}\). Add the exponents: \[-1 - 2 = -3.\]So, \(10^{-1} \cdot 10^{-2} = 10^{-3}\).
03

Simplifying the Result

The expression \(10^{-3}\) means \(\frac{1}{10^3}\). Calculate \(10^3\), which is 1000. Therefore, \(10^{-3}\) simplifies to \(\frac{1}{1000}\).

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Key Concepts

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

Rule of Exponents
The **Rule of Exponents** is a fundamental concept that helps in simplifying expressions involving exponents. When you multiply terms with the same base, you can add the exponents together to make the problem simpler. This is often remembered by the formula: if you have a base \(a\) raised to the power of \(m\) and the same base \(a\) raised to the power of \(n\), then \(a^m \cdot a^n = a^{m+n}\). For example, \(2^3 \cdot 2^2\) simplifies to \(2^{3+2} = 2^5\).
  • The rule only works when bases are the same.
  • This makes calculations faster and expressions simpler.
Understanding this rule is crucial for efficiently handling mathematical operations with exponents.
Multiplying Exponents
When diving into **multiplying exponents**, it's important to note that this process involves adding together the exponents for numbers with the same base, not multiplying the exponents themselves. Use the Rule of Exponents to simplify the task. For instance, in our exercise \(10^{-1} \cdot 10^{-2}\), both terms have the base 10, so the exponents are added:
  • \(-1 + (-2) = -3\).
After adding the exponents, the expression is simplified to \(10^{-3}\).
Keep these points in mind:
  • Make sure bases match before applying the rule.
  • Be careful with negative exponents; they signify division, not just negatives.
Simplifying Expressions with Exponents
**Simplifying expressions with exponents** often involves rewriting exponentiated terms for easier calculation, especially if they include negative exponents. A negative exponent indicates a reciprocal. For our expression \(10^{-3}\), this translates into \(\frac{1}{10^3}\).
  • Calculate \(10^3\), which equals 1000.
  • Thus, \(10^{-3}\) simplifies to \(\frac{1}{1000}\).
Whenever dealing with negative exponents:
  • Recognize they indicate an inverse operation.
  • Convert them to fractions to simplify further computation.
Simplifying expressions in this way helps in understanding and solving complex mathematical problems effectively.

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Most popular questions from this chapter

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{3}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{3}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt[3]{8}}{\sqrt[4]{4}} $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.0037\)

Simplify each of the following. Express final results using positive exponents only. $$ \frac{56 a^{\frac{1}{6}}}{8 a^{\frac{1}{4}}} $$

Avogadro's number, \(602,000,000,000,000,000,000,000\), is the number of atoms in 1 mole of a substance. Express this number in scientific notation.
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