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Chapter 10: Problem 28

Simplify. $$ \sqrt{0.0016} $$

Short Answer

Expert verified
The simplified form of \(\sqrt{0.0016}\) is approximately 0.04018.

Step by step solution

01

- Express the Number in Scientific Notation

First, express the number under the square root in scientific notation. \[ 0.0016 = 1.6 \times 10^{-3} \]
02

- Use the Property of Square Roots

Apply the property of square roots which states \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]. This gives: \[ \sqrt{1.6 \times 10^{-3}} = \sqrt{1.6} \times \sqrt{10^{-3}} \]
03

- Simplify the Square Root of the Exponent

The square root of a power of ten is simplified as follows: \[ \sqrt{10^{-3}} = 10^{-3/2} = 10^{-1.5} \]
04

- Simplify the Square Root of 1.6

Using a calculator, simplify \(\sqrt{1.6}\). This gives approximately: \[ \sqrt{1.6} \approx 1.2649 \]
05

- Combine the Results

Combine the two results: \[ 1.2649 \times 10^{-1.5} \]
06

- Convert to Decimal Form

Convert back to decimal form for the final answer. This results in: \[ 1.2649 \times 10^{-1.5} = 1.2649 \times 0.03162 \] which gives approximately: \[ 0.04018 \]

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Key Concepts

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

Scientific Notation
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in decimal form. It's commonly used in scientific calculations. In scientific notation, numbers are expressed as the product of a number (between 1 and 10) and a power of ten. For example, the number 0.0016 can be expressed as \( 1.6 \times 10^{-3} \). This form makes it easier to work with very large or very small numbers by focusing on the digits and the decimal movement.
Square Roots
The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). The square root symbol is \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \). When dealing with square roots of numbers in scientific notation, you can separate the base and the exponent for simplification. For instance, \( \sqrt{1.6 \times 10^{-3}} \) can be broken down into \( \sqrt{1.6} \times \sqrt{10^{-3}} \).
Powers of Ten
Powers of ten are a way to express very large or very small numbers. They are written as \( 10 \) raised to an exponent. The exponent shows how many times ten is multiplied by itself.
For example:
\

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