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Chapter 10: Problem 38

Solve. $$2(1-x)^{1 / 3}=4^{1 / 3}$$

Short Answer

Expert verified
The solution is x = -1.

Step by step solution

01

- Simplify the equation

First, isolate the term involving the variable. Divide both sides of the equation by 2: \[ (1-x)^{1/3} = 2^{1/3} \]
02

- Eliminate the exponent

Raise both sides of the equation to the power of 3 to eliminate the fractional exponent: \[ ( (1-x)^{1/3} )^3 = (2^{1/3})^3 \] which simplifies to \[ 1-x = 2 \]
03

- Solve for x

Isolate x on one side of the equation by subtracting 1 from both sides: \[ -x = 1 \] To make x positive, multiply both sides by -1: \[ x = -1 \]

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Key Concepts

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

fractional exponents
Fractional exponents can be tricky, but they follow the same rules as integer exponents. A fractional exponent like \((a^{1/n})\) is the same as taking the n-th root of a, written as \(\sqrt[n]{a}\).
You can break down fractional exponents into two parts:
  • The numerator (top part) of the fraction still acts like a regular exponent
  • The denominator (bottom part) of the fraction tells you the root

For example, in the equation \((1-x)^{1/3} = 2^{1/3}\), the expression \((1-x)^{1/3}\) means that you are taking the cube root of \(1-x\). Understanding this helps you make sense of the operations needed to solve the equation, particularly when you later need to 'undo' the root by raising both sides to a power.
isolating variables
Isolating the variable is a key step in solving any algebraic equation. It means getting the variable you are solving for by itself on one side of the equation.
Here are some tips for isolating variables:
  • First, perform inverse operations. If something is added to the variable, subtract it from both sides. If it's multiplied, divide both sides by that number.
  • Always do the same operation to both sides of the equation to keep it balanced

In our example, starting with \(2(1-x)^{1/3} = 4^{1/3}\), divide both sides by 2 to isolate the term with the variable:\((1-x)^{1/3} = 2^{1/3}\).
This step simplifies the equation and brings you closer to having \(x\) by itself.
raising to a power
Raising both sides of an equation to a power is a great way to eliminate fractional exponents. This step reverses the root taken by the fractional exponent.
Here's what to consider when raising to a power:
  • Make sure you raise both sides to the same power to maintain equality.
  • Be careful with negative numbers and fractional exponents, as they can introduce extraneous solutions.

Moving back to our example, after isolating the term with the variable as \((1-x)^{1/3} = 2^{1/3}\) we can eliminate the cube root by raising both sides to the power of 3: \[( (1-x)^{1/3} )^3 = (2^{1/3})^3\].
This simplifies the equation to \(1-x = 2\), making it easy to solve for \(x\).

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Most popular questions from this chapter

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[3]{(2 x+1)^{2}}}{\sqrt[5]{(2 x+1)^{2}}}$$

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{-80 a^{14}} $$

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[5]{x^{13} y^{8} z^{17}} $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$\sqrt{6} \sqrt{3}$$

Find a simplified form for \(f(x) .\) Assumex \(\geq 0\). $$\begin{aligned}&f(x)=\sqrt{20 x^{2}+4 x^{3}}-3 x \sqrt{45+9 x}+\\\&\sqrt{5 x^{2}+x^{3}}\end{aligned}$$
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