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Chapter 7: Problem 1

In Exercises \(1-21,\) solve the equation for the variable. $$ x^{3}=50 $$

Short Answer

Expert verified
Answer: The value of x in the equation \(x^3 = 50\) is approximately 3.684.

Step by step solution

01

Understand the equation

The given equation is \(x^3 = 50\). We need to solve for x by finding the value that, when cubed, results in 50.
02

Find the cube root

To solve for x, we need to find the cube root of 50. The cube root of a number is the value that, when multiplied by itself three times, gives that number. We can represent the cube root using the "鈭�" symbol or as a fractional exponent. Thus, we have two equivalent expressions for calculating the cube root of 50: $$ x = \sqrt[3]{50} $$ or $$ x = 50^{\frac{1}{3}} $$
03

Calculate the cube root

Use a calculator to find the cube root of 50: $$ x = \sqrt[3]{50} \approx 3.684 $$ or $$ x = 50^{\frac{1}{3}} \approx 3.684 $$
04

Write the final solution

Since the cube root of 50 cannot be simplified further, the final solution is: $$ x \approx 3.684 $$

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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 of a number is essential when you want to undo a cubed number. Think of it like finding the opposite of cubing a number. If you take any number and multiply it by itself twice (meaning three times in total), you will get a bigger number. However, if you start with the bigger number and want to get back to your original number, you would take the cube root.
For instance:
  • The cube root of 27 is 3 because 3 脳 3 脳 3 = 27.
  • You can express the cube root using the symbol 鈭�. For our example, it's written as 鈭�27 = 3.
But what if your number isn't a nice, neat cube like 27? That's where we use calculators. As in our exercise, the number 50 isn鈥檛 a neat cube like 27, so we use the expression \(x = 50^{\frac{1}{3}}\) to determine the cube root which is approximately 3.684.
Exponents
Exponents are used to denote the number of times a number is multiplied by itself. In the context of our equation, we start with the expression \(x^3 = 50\). The "3" here is the exponent, indicating that "x" is being multiplied by itself twice more (a total of three times).
Exponents are powerful tools:
  • They simplify expressions that would otherwise require lengthy multiplication.
  • You can use fractional exponents to express roots; for instance, \(x^{\frac{1}{3}}\) represents the cube root of x.

In our exercise, moving from \(x^3 = 50\) to finding \(x = 50^{\frac{1}{3}}\) showcases changing from a whole number exponent to a fractional one to help us solve the equation.
Algebraic Solutions
Algebraic solutions involve manipulating an equation to find the value of a variable. Solving equations allows us to find unknowns using logical operations and principles. In the case of \(x^3 = 50\), we aim to isolate "x" and find its value. Here's how the process unfolds:
  • The task: Find "x" such that when cubed, it equals 50.
  • Step: Recognize that taking the cube root will isolate x. Thus, x is expressed as \(\sqrt[3]{50}\) or \(50^{\frac{1}{3}}\).
  • Outcome: Use a calculator for precision, resulting in \(x \approx 3.684\).
Through this process, we logically work through the equation to arrive at a solution that makes sense numerically and algebraically. Always remember, understanding the operations and manipulations is crucial to solving algebraic equations effectively.

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Most popular questions from this chapter

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{1 / 2}=12 $$

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

If \(z\) is proportional to a power of \(x\) and \(y\) is proportional to the same power of \(x\), is \(z+y\) proportional to a power of \(x ?\)

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The circulation time, \(T,\) of a mammal as a function of its body mass, \(B\) : $$ T=M \sqrt[4]{B}. $$

The surface area, \(S\), in \(\mathrm{cm}^{2}\), of a mammal of mass \(M\) \(\mathrm{kg}\) is given by \(S=k M^{2 / 3},\) where \(k\) depends on the body shape of the mammal. For people, assume that \(k=1095 .\) (a) Find the body mass of a person whose surface area is \(21,000 \mathrm{~cm}^{2}\) (b) What does the solution to the equation \(1095 M^{2 / 3}=30,000\) represent? (c) Express \(M\) in terms of \(S\).
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