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Chapter 10: Problem 41

Solve each equation. $$ \sqrt[3]{2 m-1}=\sqrt[3]{m+13} $$

Short Answer

Expert verified
m = 14

Step by step solution

01

Equalize the Radicals

Since both sides of the equation are cubic roots, set the radicands equal to each other as the next step. So, write: \[ 2m - 1 = m + 13 \]
02

Isolate Variable Terms

Subtract \(m\) from both sides of the equation to isolate the terms involving \(m\): \[ 2m - m - 1 = m - m + 13 \]This simplifies to: \[ m - 1 = 13 \]
03

Solve for \(m\)

Add 1 to both sides of the equation to solve for \(m\): \[ m - 1 + 1 = 13 + 1 \]This simplifies to: \[ m = 14 \]
04

Verify the Solution

Substitute \(m = 14\) back into the original equation to verify: \[ \sqrt[3]{2(14) - 1} = \sqrt[3]{14 + 13} \]Simplifying both sides: \[ \sqrt[3]{28 - 1} = \sqrt[3]{27} \]\[ \sqrt[3]{27} = \sqrt[3]{27} \]Both sides are equal, confirming that \(m = 14\) is correct.

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Key Concepts

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

Cubic Roots
A cubic root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cubic root of 27 is 3 because: \(3 \times 3 \times 3 = 27\).
  • In mathematical notation, we write the cubic root of a number as \sqrt[3]{x}\.
  • Cubic roots are similar to square roots, but they work with powers of three instead of two.
In the initial equation we encountered, \sqrt[3]{2m -1} = \sqrt[3]{m +13}\, both sides are expressed in terms of cubic roots. This allows us to set the expressions inside the cubic roots equal to each other because the cubic root function on both sides will yield the same result.
Isolating Variables
Isolating a variable means rearranging the equation so that the variable of interest stands alone on one side of the equation. This is a crucial step in solving equations.
Let's break down how this works with our given equation:
  • The original equation after equalizing the radicands is \2m - 1 = m + 13\.
  • We want to get all the terms involving the variable \(m\) on one side. Subtract \(m\) from both sides to simplify: \(2m - m - 1 = m - m + 13\).
  • This results in \(m - 1 = 13\).
By isolating the variable \(m\), we can easily solve for its value.
Equation Verification
Verification is the process of checking whether a solution to an equation is correct by substituting the obtained solution back into the original equation.
In our example, the final value obtained for \(m\) was 14. To verify this:
  • Substitute \(m = 14\) back into the original equation: \sqrt[3]{2(14) - 1} = \sqrt[3]{14 + 13}\.
  • This simplifies to: \sqrt[3]{27} = \sqrt[3]{27}\.
  • Since both sides are equal, the solution is confirmed to be correct.
Verification is an essential step as it ensures that you haven't made any errors during the solution process.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations and properties to simplify and solve equations. Here's a summary of the steps used in this problem:
  • First, we set the radicands (the expressions inside the cubic roots) equal to each other: \(2m - 1 = m + 13\).
  • Next, we isolated the variable \(m\) by subtracting \(m\) from both sides: \(m - 1 = 13\).
  • Then, we solved for \(m\) by adding 1 to both sides: \(m = 14\).
  • Finally, we verified the solution by plugging it back into the original equation.
Mastering algebraic manipulation is key to efficiently solving complex equations and allows you to understand the relationships between variables.

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