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Chapter 3: Problem 44

Solve. $$\sqrt{m+1}-5=8$$

Short Answer

Expert verified
The value of \(\textstyle m\) is 168.

Step by step solution

01

Isolate the Square Root

To start, isolate the square root term on one side of the equation. The equation is \(\textstyle \sqrt{m+1} - 5 = 8\). Add 5 to both sides to get: \(\textstyle \sqrt{m+1} = 13\).
02

Remove the Square Root

Square both sides of the equation to eliminate the square root. This gives: \(\textstyle (\sqrt{m+1})^2 = 13^2\), which simplifies to: \(\textstyle m+1 = 169\).
03

Solve for m

Subtract 1 from both sides to find the value of \(\textstyle m\). This gives: \(\textstyle m = 168\).
04

Verify the Solution

Substitute \(\textstyle m = 168\) back into the original equation to ensure it is correct. The original equation is \(\textstyle \sqrt{168+1} - 5 = 8\). Simplifying inside the square root gives: \(\textstyle \sqrt{169} - 5 = 8\), which simplifies to: \(\textstyle 13 - 5 = 8\), confirming that the solution is correct.

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Key Concepts

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

Isolating the square root
The first step to solving equations involving square roots is to isolate the square root term. Let's say you have an equation like \(\textstyle \sqrt{m+1} - 5 = 8\). Your goal is to get the square root by itself on one side. To do this, add 5 to both sides of the equation, which gives: \(\textstyle \sqrt{m+1} = 13\). By isolating the square root, you make it easier to proceed with solving the equation. Always remember that isolating the square root simplifies the steps that follow.
Squaring both sides
Once the square root term is isolated, you need to eliminate the square root to solve for the variable. This is done by squaring both sides of the equation. For our example, starting with \(\textstyle \sqrt{m+1} = 13\), we square both sides to get: \(\textstyle (\sqrt{m+1})^2 = 13^2\). This removes the square root, leaving: \(\textstyle m+1 = 169\). Squaring both sides is a powerful technique but must be applied correctly to avoid errors.
Verifying solutions in algebra
Verifying your solutions is crucial in algebra to ensure that your solution is correct and satisfies the original equation. After finding \(\textstyle m = 168\), substitute it back into the original equation to check your work. The original equation is \(\textstyle \sqrt{168+1} - 5 = 8\). Simplify inside the square root to get: \(\textstyle \sqrt{169} - 5 = 8\), which simplifies further to: \(\textstyle 13 - 5 = 8\). Since both sides of the equation are equal, the solution \(\textstyle m = 168\) is verified. Always substitute your solution back into the original equation to confirm its validity.

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