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Chapter 10: Problem 52

For Exercises \(49-64\), convert to standard form. $$2 \times 10^{-5}$$

Short Answer

Expert verified
0.00002

Step by step solution

01

Identify the Scientific Notation

The given number is in scientific notation, which is written as: \(2 \times 10^{-5}\). Here, the base is 10 and the exponent is -5.
02

Understand the Negative Exponent

A negative exponent means moving the decimal point to the left. In this case, we need to move the decimal point 5 places to the left.
03

Move the Decimal Point

Start with the number 2 and move the decimal point 5 places to the left. So, it becomes: \(0.00002\).
04

Convert to Standard Form

After moving the decimal point, the number in standard form is: \(0.00002\).

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Key Concepts

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

Negative Exponents
Negative exponents can be a bit tricky at first, but they follow a simple rule. A negative exponent means that you will be moving the decimal point to the left instead of the right. For instance, in the problem, you started with: - $$2 \times 10^{-5}$$ - Here, the base is 10, and the exponent is -5. The negative sign tells us we're dealing with a number smaller than 1. When you have a negative exponent, you're essentially dividing by that number of 10s. For example, $$10^{-1} = \frac{1}{10} = 0.1$$ and $$10^{-2} = \frac{1}{10^{2}} = 0.01$$. For every step left, the number gets ten times smaller.
Decimal Point Movement
When converting from scientific notation to standard form, moving the decimal point is a critical step. To convert $$2 \times 10^{-5}$$ into standard form, you need a solid grasp of decimal point movement. The exponent of -5 means that you will move the decimal point 5 places to the left. Here's how you can visualize it:
  • Start with the number 2.
  • Imagine the decimal point at the end (2.0).
  • Move the decimal point 5 places to the left.
  • Fill any empty spaces with zeros.
So, you move it one place to get 0.2, a second time to get 0.02, a third time to get 0.002, fourth for 0.0002, and finally to 0.00002. Therefore, $$2 \times 10^{-5} = 0.00002$$. By practicing moving the decimal point, you鈥檒l get better at converting between scientific notation and standard form.
Standard Form Conversion
Standard form is the typical way of writing numbers without using exponents. To convert a number from scientific notation to standard form, simply move the decimal point as indicated by the exponent. In the given exercise, we had $$2 \times 10^{-5}$$, and by moving the decimal point 5 places to the left, we changed it to the standard form $$0.00002$$. Here are a few quick tips to remember:
  • If the exponent is positive, move the decimal point to the right.
  • If the exponent is negative, move the decimal point to the left.
  • For each move, add zeros to fill in the gaps as needed.
Practicing these steps will make it easier to understand and perform conversions between scientific notation and standard form.

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