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Chapter 5: Problem 70

\(\left(8 \times 10^{5} \mathrm{~m}\right)^{3}\)

Short Answer

Expert verified
512 \times 10^{15} \text{ m}^3

Step by step solution

01

Understand the Expression

The given expression is \(\big(8 \times 10^{5} \text{ m}\big)^3\). It involves cubing a product.
02

Apply Exponent Rules

When cubing a product, use the rule \( (a \times b)^n = a^n \times b^n \). Applying this, the expression becomes \(\big(8 \times 10^{5}\big)^3 = 8^3 \times (10^{5})^3\).
03

Calculate Individual Cubes

Calculate \(8^3\) and \((10^{5})^3\). First, \( 8^3 = 8 \times 8 \times 8 = 512 \). Next, use the power of a power rule for exponents: \((10^5)^3 = 10^{5 \times 3} = 10^{15}\).
04

Combine the Results

Combining the calculated values gives us: \( 512 \times 10^{15} \text{ m}^3 \).

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Key Concepts

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

Cubing Numbers
Cubing a number means raising it to the power of three. In simpler terms, you multiply the number by itself three times. For example, to cube the number 2, you would calculate:
2 脳 2 脳 2 = 8.
This rule applies to any number or variable you are cubing. In our exercise, we needed to cube the number 8. By following the same rule, we get:
8 脳 8 脳 8 = 512.
Understanding cubing helps simplify complex expressions in algebra.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number you are multiplying, while the exponent tells you how many times to multiply the base by itself. For example, in the expression 3^4 (read as 'three to the fourth power'), the base is 3 and the exponent is 4, which means:
3 脳 3 脳 3 脳 3 = 81.
In our exercise, we applied the exponentiation rule to products:
\( (a \times b)^n = a^n \times b^n \).
This allowed us to separate the expression into manageable parts. Understanding exponentiation is crucial for both basic and advanced algebraic operations.
Scientific Notation
Scientific notation is a way to express very large or very small numbers in a compact form. It is especially useful in mathematics, science, and engineering. The format is:
\( a \times 10^n \),
where 'a' is a number between 1 and 10, and 'n' is an integer. For example, 3000 is written as 3 脳 10^3 in scientific notation.
In our exercise, we used scientific notation to handle large exponents easily. The initial expression had 10 raised to the power of 5. When cubed, it became: (10^5)^3 = 10^{5 \times 3} = 10^{15}.
Knowing how to use scientific notation simplifies calculations and helps in understanding the scale of numbers.

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Most popular questions from this chapter

\(\left(2 x^{2}-11 x-21\right) \div(x-7)\)

\(\left(r^{3}\right)^{5}\)

The length of a rectangle is \(8 \mathrm{in}\). longer than the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

\(\left(w^{3}-w-5\right)-\left(w^{3}-w^{2}-5\right)\)

The prefix tri means "three." State an English word besides trinomial that includes the prefix tri. Explain the meaning of the word.
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