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Chapter 5: Problem 17

Simplify each numerical expression. \(10^{-5} \cdot 10^{2}\)

Short Answer

Expert verified
The simplified expression is \(10^{-3}\).

Step by step solution

01

Understand the Problem

We need to simplify the expression \(10^{-5} \cdot 10^{2}\). This involves working with exponents to combine the terms into a simpler form.
02

Apply the Product of Powers Rule

The product of powers rule states that when you multiply two expressions with the same base, you can add the exponents. Therefore, apply the rule here: \(10^{-5} \cdot 10^{2} = 10^{-5+2}\).
03

Simplify the Exponents

Add the exponents: \(-5 + 2 = -3\). Therefore, \(10^{-5+2} = 10^{-3}\).
04

Write the Final Simplified Expression

The simplified form of the expression is \(10^{-3}\). Therefore, \(10^{-5} \cdot 10^{2} = 10^{-3}\).

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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 working with exponents, the product of powers rule is incredibly handy, especially when you're multiplying numbers that have the same base.
The rule states that when you have two numbers with the same base and you multiply them, you can simply add their exponents together. This simplifies the process of multiplication when dealing with powers.
For instance, with the expression \(10^{-5} \cdot 10^{2}\), both numbers share the same base of 10.
  • According to the product of powers rule, you take the exponents, \(-5\) and \(2\), and add them together.
  • This gives you \(-5 + 2 = -3\), resulting in a simplified expression: \(10^{-3}\).
This technique helps to swiftly simplify complex expressions with like bases, making calculations much easier for students.
Simplifying Expressions
Simplifying expressions involves combining the elements of an expression into a more concise form without changing its value.
It's all about making things easier to understand and work with.
  • Start by identifying the parts of the expression that can be combined or simplified, such as like bases in exponents.
  • Use rules like the product of powers rule to help in reducing the complexity of the terms. In the example \(10^{-5} \cdot 10^{2}\), we utilize the rule to add the exponents.
  • This results in a much simpler expression: \(10^{-3}\).
By continually practicing how to simplify, students can improve their mathematical problem-solving skills and fluidly handle more challenging equations with ease.
Negative Exponents
Negative exponents might seem intimidating at first, but they're quite straightforward once you understand them.
Essentially, a negative exponent signifies the reciprocal of that base raised to the corresponding positive exponent.
  • For instance, \(10^{-3}\) can be interpreted as \(\frac{1}{10^{3}}\).
  • This means moving the base and the positive exponent to the denominator of a fraction with 1 as the numerator.
  • So, instead of multiplying, you'll be dividing, showing how negative exponents turn multiplication into division.
Understanding negative exponents is crucial because it extends the concept of powers, turning what seems to be a complex situation into a more manageable process, providing clarity in various mathematical problems.

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Most popular questions from this chapter

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ \frac{(0.0045)(60,000)}{(1800)(0.00015)} $$

Simplify each of the following. Express final results using positive exponents only. $$ \left(3 x^{\frac{1}{4}} y^{\frac{1}{5}}\right)^{3} $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.347\)

For Problems \(1-18\), write each of the following in scientific notation. \(0.00000000003\)

For Problems \(19-32\), write each of the following in ordinary decimal notation. $$ (6)\left(10^{-9}\right) $$
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