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Chapter 9: Problem 36

Divide as indicated. Write your answer using only positive exponents. $$10^{-4} \div 10^{2}$$

Short Answer

Expert verified
The answer is \( \frac{1}{10^6} \).

Step by step solution

01

Understand the Division Rule for Exponents

When dividing expressions with the same base, we subtract the exponents. The division rule for exponents states that \( a^m \div a^n = a^{m-n} \), where \(a\) is the base, and \(m\) and \(n\) are the exponents.
02

Apply the Rule to the Problem

Apply the exponent rule to the problem \(10^{-4} \div 10^{2}\). This results in \(10^{-4-2}\).
03

Simplify the Exponent

Simplify by subtracting the exponents: \(-4 - 2 = -6\). Thus, the expression becomes \(10^{-6}\).
04

Convert to Positive Exponents

To express the result with positive exponents, recall that \(a^{-b} = \frac{1}{a^b}\). Therefore, \(10^{-6} = \frac{1}{10^6}\).

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Key Concepts

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

Exponent Rules
Exponent rules are essential for simplifying expressions involving powers. When you divide two expressions that have the same base, you can use the division rule of exponents. This rule states that you subtract the exponent of the divisor from the exponent of the dividend.

Here's the formula to remember:
  • For any base \(a\), \(a^m \div a^n = a^{m-n}\).
In simple terms, if you're dividing two powers with the same base, just subtract the exponent in the denominator from the exponent in the numerator.

Let's take an example: If you have \(10^5 \div 10^3\), you subtract 3 from 5, resulting in \(10^{5-3}\), which simplifies to \(10^2\). This rule makes it easier to handle large numbers or when comparing powers.
Negative Exponents
Negative exponents can initially seem tricky, but they follow straightforward rules. A negative exponent implies that the base is on the wrong side of a fraction line, so you just "flip" it to the other side.

Keep this in mind:
  • \(a^{-n} = \frac{1}{a^n}\).
This means a negative exponent indicates a reciprocal. For example, \(10^{-3}\) is the same as \(\frac{1}{10^3}\).

Using negative exponents helps express very small numbers in scientific notation. When you convert a negative exponent into a positive one, you're essentially making it easier to understand and use in further calculations.
Positive Exponents
Once you've dealt with negative exponents, converting them to positive exponents is a crucial step. Positive exponents make it easier to grasp the value of a number because they suggest repeated multiplication of the base.

Remember these points:
  • Positive exponents reflect standard multiplication.
  • \(a^n\) means multiplying \(a\) by itself \(n\) times.
For example, \(10^3\) implies \(10 \times 10 \times 10\), which equals 1000.

In solving exercises, such as the original problem of \(10^{-4} \div 10^2\), reaching the last step where you express the solution with positive exponents, like \(\frac{1}{10^6}\), is vital. It helps finalize the expression neatly, making it comprehensible for others.

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Most popular questions from this chapter

Simplify the following expressions. \(\left(9 a^{5} b^{6}\right)^{2}\left(8 a^{5} b\right)\)

Simplify the following expressions. \(\left(9 x^{4}\right)^{2}\)

Multiply the following expressions. $$(7 x-3)(7 x+3)$$

Multiply the following expressions. $$4 x^{2} y^{3}\left(3 x^{4}+2 y^{3}\right)$$

The Great Pyramid at Giza is one of the largest and oldest man-made structures in the world. It weighs over \(10^{10}\) kilograms. If each stone making up the pyramid weighs approximately \(4,000\) kilograms, how many stones make up the structure? Write your answer in scientific notation.
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