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Chapter 0: Problem 131

Use a calculator to approximate each radical. Round your answer to two decimal places. $$\frac{3 \sqrt[3]{5}-\sqrt{2}}{\sqrt{3}}$$

Short Answer

Expert verified
Approximately 2.15

Step by step solution

01

- Evaluate the cube root

First, calculate the approximate value of the cube root of 5 using a calculator. \[ \sqrt[3]{5} \approx 1.71 \]
02

- Calculate the square root

Next, find the approximate value of the square root of 2. \[ \sqrt{2} \approx 1.41 \]
03

- Substitute the values

Substitute the calculated values back into the expression. \[ 3 \sqrt[3]{5} - \sqrt{2} \approx 3 \times 1.71 - 1.41 = 5.13 - 1.41 \]
04

- Simplify the numerator

Simplify the expression found in the previous step. \[ 5.13 - 1.41 = 3.72 \]
05

- Evaluate the square root in the denominator

Calculate the square root of 3. \[ \sqrt{3} \approx 1.73 \]
06

- Divide the numerator by the denominator

Finally, divide the simplified numerator by the approximation of the square root of 3. \[ \frac{3.72}{1.73} \approx 2.15 \]

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Key Concepts

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

Cube Root Calculation
When calculating cube roots, you're finding a number that, when multiplied by itself three times, gives you the original number. For example, with \(\text{consider the expression} \sqrt[3]{8}\), since 2 \(2 \times 2 \times 2 = 8\), this makes 2 the cube root of 8.
For more complicated numbers, calculators can help. Take the cube root of 5, expressed mathematically as \(\text{as} \sqrt[3]{5}\), we approximate it as 1.71 using a calculator.
Here's the general approach:
  • Input your number into the calculator.
  • Use the cube root function (often depicted as \( \sqrt[3]{ } \) ).
  • Read the result to the desired decimal places.
This step makes complex algebraic expressions manageable.
Square Root Calculation
Calculating square roots helps find a number that, when multiplied by itself, equals the original number. For instance, the square root of 4 is 2 because \( \2 \times 2 = 4 \). If the number isn't a perfect square, a decimal approximation is necessary. To approximate the square root of 2:
  • Use a calculator.
  • Select the square root function (usually \( \sqrt{} \)).
  • Enter the number and read the approximation.
For example, \( \sqrt{2} \approx 1.41\). Another example is finding \( \sqrt{3} \approx 1.73\). Breaking down problems into smaller steps helps manage calculations of these roots within larger expressions.
Numerical Approximation
Numerical approximation is the process of finding an approximate numerical value for a mathematical expression. This technique is vital when working with irrational numbers or complex expressions where exact values are hard to obtain. Consider the expression \( \frac{3 \sqrt[3]{5} - \sqrt{2}}{\sqrt{3}}\). This involves:
  • Approximating \( \sqrt[3]{5}\) as 1.71.
  • Finding \( \sqrt{2} \approx 1.41 \).
  • Combining these values into the expression to simplify the numerator.
  • Dividing by \( \sqrt{3}\) approximated as 1.73.
The result is \( \frac{3.72}{1.73}\) which we further approximate to 2.15. This method helps simplify otherwise challenging expressions and enhances problem-solving efficiency.
Basic Algebra Operations
Basic algebra operations such as addition, subtraction, multiplication, and division are foundational for solving complex equations. In the expression \( \frac{3 \sqrt[3]{5} - \sqrt{2}}{\sqrt{3}}\), we:
  • Multiply \( \sqrt[3]{5} \) by 3 to get 3 \times 1.71 = 5.13.
  • Subtract \( \sqrt{2} \approx 1.41 \) from this product, giving us 5.13 - 1.41 = 3.72.
These steps demonstrate how approximations and basic arithmetic allow us to simplify and solve algebraic expressions. Understanding when and how to use these operations is crucial for tackling more advanced math problems.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$2 x(3 x+4)^{4 / 3}+x^{2} \cdot 4(3 x+4)^{1 / 3}$$

Simplify each expression. Assume that all variables are positive when they appear. $$3 x \sqrt{9 v}+4 \sqrt{25 v}$$

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

Simplify each expression. Assume that all variables are positive when they appear. $$5 \sqrt[3]{2}-2 \sqrt[3]{54}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$
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