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Chapter 0: Problem 6

Calculate \(\left(\sqrt[3]{8.15 \times 10^{4}}-1.32\right)^{2} / 3.24\)

Short Answer

Expert verified
The result is approximately 562.19.

Step by step solution

01

Calculate Cube Root

We first need to compute the cube root of the number inside the parentheses. The expression given is \( \sqrt[3]{8.15 \times 10^4} \). Calculate this to get approximately 43.99.
02

Subtract 1.32

Subtract 1.32 from the result obtained in Step 1. Thus, \( 43.99 - 1.32 = 42.67 \).
03

Square the Result

Now, square the result from Step 2. That means you compute \( 42.67^2 \), which equals approximately 1821.51.
04

Divide by 3.24

Finally, divide the squared result by 3.24 as shown in the expression. So, \( \frac{1821.51}{3.24} \approx 562.19 \).

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Key Concepts

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

Cube Root
The cube root is the value that, when used in multiplication three times, gives the original number. It is represented by the radical sign with a small three, such as \( \sqrt[3]{x} \). When calculating the cube root of a number like \( \sqrt[3]{8.15 \times 10^{4}} \), you're looking for a number that, when multiplied by itself twice more, will equal \( 8.15 \times 10^{4} \).
  • One method to find cube roots is by trial and error, especially for larger numbers like \( 8.15 \times 10^{4} \).
  • Since cube roots involve estimating, using a calculator is a common approach, as it can provide the precise cube root more swiftly, yielding approximately 43.99 in this case.
  • Understanding cube roots is crucial in simplifying expressions and solving equations involving cubic terms.
Exponentiation
Exponentiation is a mathematical operation involving numbers or expressions being raised to a power. In simple words, it is repeated multiplication of a number by itself a specified number of times. When we compute \( a^n \), \( a \) is the base, and \( n \) is the exponent.
  • For example, \( (42.67)^2 \) means multiplying 42.67 by itself once, resulting in approximately 1821.51.
  • Exponentiation is a critical concept, prominently used in calculations involving growth rates, scientific computations, and more.
  • It helps in solving exponential equations and finding geometric progressions efficiently.
  • Squaring and cubing are the most common forms of exponentiation used in day-to-day problems.
Arithmetic Operations
Arithmetic operations are the building blocks of mathematics, consisting of addition, subtraction, multiplication, and division. They are used to solve equations and perform calculations that range from simple to complex.
  • In the original problem, each operation was used systematically to solve the expression.
  • Subtraction occurs first: \( 43.99 - 1.32 = 42.67 \), simplifying the initial value before applying more complex operations.
  • The next operation is exponentiation where \( 42.67 \) is squared.
  • Finally, division simplifies the squared value: \( \frac{1821.51}{3.24} \approx 562.19 \).
  • Mastering these operations is essential because they form the basis of all mathematical problem-solving.

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Most popular questions from this chapter

Plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{array}{l} y=x-1 \\ 2 x^{2}+3 y^{2}=12 \end{array} $$

Your computer algebra system (CAS) may allow the use of parameters in defining functions. In each case, draw the graph of \(y=f(x)\) for the specified values of the parameter \(k\), using the same axes and \(-5 \leq x \leq 5\). (a) \(f(x)=|k x|^{0.7}\) for \(k=1,2,0.5,\) and 0.2 . (b) \(f(x)=|x-k|^{0.7}\) for \(k=0,2,-0.5,\) and \(-3 .\) (c) \(f(x)=|x|^{k}\) for \(k=0.4,0.7,1,\) and \(1.7 .\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(|x|+|y|=1\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x=-y^{2}+1\)

Sketch the graph of each equation. $$ x^{2}-2 x+y^{2}=3 $$
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