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Chapter 8: Problem 42

Solve equation. \(\sqrt[3]{2 k-11}=\sqrt[3]{5 k+1}\)

Short Answer

Expert verified
k = -4

Step by step solution

01

Understand the Equation

Identify that the given equation is \(\frac[3]{2 k-11}=\frac[3]{5 k+1}\). The equation has cube roots on both sides.
02

Remove the Cube Roots

To isolate the expressions under the cube roots, cube both sides of the equation: \[ \big( \frac[3]{2 k-11} \big)^3 = \big( \frac[3]{5 k+1} \big)^3 \] which simplifies to \[ 2k - 11 = 5k + 1. \]
03

Isolate the Variable

To find the value of the variable k, start by moving all the k terms to one side of the equation: \[ 2k - 5k = 1 + 11 \] which simplifies to \[ -3k = 12. \]
04

Solve for k

Divide both sides by -3 to solve for k: \[ k = \frac{12}{-3} \] which simplifies to \[ k = -4. \]

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Key Concepts

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

cube roots
Cube roots can be tricky, but with some practice, they become easier to manage. The cube root of a number x, written as \(\text{\( \sqrt[3]{x}\)}\), is a value that, when multiplied by itself three times, gives the number x.
For example, the cube root of 27 is 3 because 3 \(\times \) 3 \(\times \) 3 = 27.
In the given problem, we have cube roots on both sides of the equation: \(\text{\( \sqrt[3]{2 k - 11} = \sqrt[3]{5 k + 1}\)}\). To solve it, we first need to eliminate the cube roots by cubing both sides.
isolating variables
Once we've removed the cube roots by cubing both sides, we get: \(\text{\(2k - 11 = 5k + 1\)}\).
Our next task is to isolate the variable k.
To do this, we need to move all terms involving k to one side of the equation and the constant terms to the other side.
In our case, we'll move the k terms to the left:
\(\text{\(2k - 5k\)}\) and the constant terms to the right: \(\text{\(1 + 11\)}\).
This simplifies to: \(\text{\(-3k = 12\)}\).
simplifying equations
Simplifying equations helps to make them more manageable.
After isolating the variable terms and constants in the equation \(\text{\(-3k = 12\)}\), we further simplify it by performing arithmetic operations.
Here, we divide both sides by -3 to solve for k: \(\text{\( \frac{12}{-3} = -4\)}\).
algebraic solutions
Algebraic solutions require step-by-step manipulation of equations to find the value of variables. In our example, the key steps are:
  • First, we identify and remove cube roots.
  • Then we isolate the variable terms.
  • Next, we simplify the equation by grouping like terms.
  • Finally, we solve for the variable k by performing division.
These steps help us solve the equation \(\text{\( \sqrt[3]{2 k-11} = \sqrt[3]{5 k+1}\)}\) to find that k = -4.

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