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Chapter 0: Problem 11

Compute the numbers. $$(.000001)^{1 / 3}$$

Short Answer

Expert verified
0.01

Step by step solution

01

- Understand the Expression

The expression given is \( (.000001)^{1/3} \). It represents the cube root of 0.000001.
02

- Rewrite in Scientific Notation

Rewrite 0.000001 in scientific notation: \(0.000001 = 1 \times 10^{-6}\). So, the expression becomes \( (1 \times 10^{-6})^{1/3} \).
03

- Apply the Exponent

Applying the exponentiation rule \( (a \times 10^b)^n = a^n \times 10^{bn} \, we get: \ (1 \times 10^{-6})^{1/3} = 1^{1/3} \times 10^{(-6 \times 1/3)} \).
04

- Simplify

Simplify the terms: \( 1^{1/3} = 1 \, and \ -6 \times 1/3 = -2 \. Hence, the expression simplifies to \ 1 \times 10^{-2} \).
05

- Write the Final Answer

The simplified form \( 1 \times 10^{-2} \) converts back to decimal notation as 0.01.

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Key Concepts

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

Scientific Notation
Scientific notation helps express very large or very small numbers in a concise form, making calculations more manageable. To write a number in scientific notation:
  • Move the decimal point so that there is one non-zero digit to its left.
  • Count how many places you moved the decimal point. This number becomes the exponent of 10.
For example, 0.000001 can be written as \( 1 \times 10^{-6} \). Here, the decimal moves six places to the right, hence the exponent -6. This is the value used in the original exercise before further calculations.
Exponentiation Rules
Exponentiation involves raising numbers to a power, and this operation follows specific rules:
  • \( (a^m)^n = a^{m \times n} \): The power rule states that you multiply the exponents when raising a power to another power.
  • \( (a \times b)^n = a^n \times b^n \): The product-to-power rule states that each number in a product raised to a power should be raised to that power individually.
Applying these rules simplifies complex expressions. For instance, from \( (1 \times 10^{-6})^{1/3} \) to \( 1^{1/3} \times 10^{-6 \times 1/3} \), we apply these rules to break down and simplify the terms.
Simplifying Roots
Simplifying roots means finding simpler expressions for root values:
  • The cube root (or any root) asks what number, when multiplied by itself a specified number of times, equals the target number.
  • For \( x^{1/n} \), where n is the root, find values that satisfy the equation.
In our example, we simplify \( 1^{1/3} \) to 1 since 1 raised to any power remains 1, and calculate \( 10^{-6 \times 1/3} \), giving \( 10^{-2} \). Thus, \( 1 \times 10^{-2} \) equals 0.01, showing we simplified 0.000001 to a more digestible form using roots.

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Most popular questions from this chapter

Graph the function with the specified viewing window setting. $$f(x)=\frac{1}{x^{2}+1} ;[-4,4] \text { by }[-.5,1.5]$$

Let \(f(x)=x^{2} .\) Graph the functions \(f(x+1)\) \(f(x-1), f(x+2),\) and \(f(x-2) .\) Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x)),\) where \(g(x)=x+a\) for some constant \(a\) Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}.\)

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