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Chapter 0: Problem 112

What is the value of \((0.1)^{3} \cdot(20)^{3} ?\)

Short Answer

Expert verified
The value is 8.

Step by step solution

01

- Rewrite the Expression Using Powers

First, observe the expression \((0.1)^{3} \cdot (20)^{3}\). Notice that both terms are raised to the same power (3). This allows us to use the property \((a \cdot b)^{3} = a^{3} \cdot b^{3}\). Hence, we can rewrite the expression as \((0.1 \cdot 20)^{3}\).
02

- Simplify Inside the Parentheses

Next, multiply \(0.1\) by \(20\). \(0.1 \cdot 20 = 2\). Therefore, the expression now is \(2^{3}\).
03

- Evaluate the Power

Finally, calculate \(2^{3}\). This means multiplying \(2 \cdot 2 \cdot 2 = 8\).

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Key Concepts

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

powers and exponents
Powers and exponents are fundamental in mathematics. Exponents indicate how many times a number, known as the base, is multiplied by itself.
For example, in \(2^3\), '2' is the base and '3' is the exponent. Exponents make it easier to write and work with large numbers.
Some key properties of exponents you should know are:
  • \(a^m \times a^n = a^{m+n}\) - When multiplying like bases, add their exponents.

  • \(a^m / a^n = a^{m-n}\) - When dividing like bases, subtract the exponents.

  • \((a^m)^n = a^{mn}\) - When raising an exponent to another power, multiply the exponents.

Understanding these properties allows you to simplify and solve complex expressions involving exponents.
multiplication
Multiplication is a basic arithmetic operation that combines groups of equal size. It's fundamental for understanding more complex mathematics.
Here are some straightforward guidelines for multiplication:
  • Combine like terms: For example, \2 \times 3 = 6\.

  • Multiplying decimals: Align the numbers by their decimal points and then multiply as if they were whole numbers. Adjust the decimal point in the result accordingly.

In our original exercise, we applied these principles. First, we rewrote the expression: \(0.1^3 \times 20^3\). Then, we used the property of exponents: \((a \times b)^3 = a^3 \times b^3\). This allowed us to simplify the expression to \(2^3\), making it much easier to evaluate.
mathematical properties
Mathematical properties are rules that provide the framework for manipulating numbers and equations. Understanding these properties helps to simplify and solve equations.
Some key properties include:
  • Multiplicative Identity: Multiplying any number by 1 leaves it unchanged: \a \times 1 = a\.

  • Distributive Property: \(a(b + c) = ab + ac\) - This property helps in breaking down complex multiplication problems.

  • Associative Property: \(a(bc) = (ab)c\) - The way numbers are grouped in multiplication doesn't change the result.
In our exercise, the associative property was indirectly used when we rewrote the initial expression \(0.1^3 \times 20^3\) into \((0.1 \times 20)^3 = 2^3\). These properties are essential for simplifying steps and ensuring accurate calculations.

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Most popular questions from this chapter

Simplify each expression. $$(-1000)^{-1 / 3}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\left(9-x^{2}\right)^{1 / 2}+x^{2}\left(9-x^{2}\right)^{-1 / 2}}{9-x^{2}} \quad-3

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x}-2}{x-4} x \neq 4$$

Simplify each expression. $$(-64)^{1 / 3}$$

The period \(T\), in seconds, of a pendulum of length \(l,\) in feet, may be approximated using the formula $$T=2 \pi \sqrt{\frac{l}{32}}$$ Express your answer both as a square root and as a decimal approximation. Find the period \(T\) of a pendulum whose length is 64 feet.
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