To solve the equation [tex]\(\sqrt{m} - \sqrt{m - \sqrt{81}} = 1\)[/tex], let us go through the steps methodically:
1. Understand the Components:
This equation has two square roots: [tex]\(\sqrt{m}\)[/tex] and [tex]\(\sqrt{m - \sqrt{81}}\)[/tex]. Recognize that [tex]\(\sqrt{81} = 9\)[/tex].
2. Substitute and Simplify:
Substitute [tex]\(\sqrt{81}\)[/tex] with [tex]\(9\)[/tex]:
[tex]\[\sqrt{m} - \sqrt{m - 9} = 1\][/tex]
3. Isolate One of the Square Roots:
Let's isolate [tex]\(\sqrt{m - 9}\)[/tex] on one side of the equation.
[tex]\[\sqrt{m} - 1 = \sqrt{m - 9}\][/tex]
4. Square Both Sides to Eliminate the Square Roots:
Squaring both sides to remove the roots:
[tex]\[(\sqrt{m} - 1)^2 = (\sqrt{m - 9})^2\][/tex]
Expanding the left side:
[tex]\[\sqrt{m}^2 - 2\sqrt{m} \cdot 1 + 1^2 = m - 9\][/tex]
Simplifying:
[tex]\[m - 2\sqrt{m} + 1 = m - 9\][/tex]
5. Isolate the Radical Term:
Subtract [tex]\(m\)[/tex] from both sides:
[tex]\[-2\sqrt{m} + 1 = -9\][/tex]
Simplify further:
[tex]\[-2\sqrt{m} = -10\][/tex]
6. Solve for [tex]\(\sqrt{m}\)[/tex]:
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[\sqrt{m} = 5\][/tex]
7. Square Both Sides Again:
Square both sides to solve for [tex]\(m\)[/tex]:
[tex]\[m = 25\][/tex]
So, the solution to the equation [tex]\(\sqrt{m} - \sqrt{m - 9} = 1\)[/tex] is:
[tex]\[m = 25\][/tex]
To verify, substitute [tex]\(m = 25\)[/tex] back into the original equation to ensure it holds true:
[tex]\[\sqrt{25} - \sqrt{25 - \sqrt{81}} = 5 - \sqrt{25 - 9} = 5 - \sqrt{16} = 5 - 4 = 1\][/tex]
Since the original equation is satisfied, the solution [tex]\(m = 25\)[/tex] is correct. Thus, the value of [tex]\(m\)[/tex] is:
[tex]\[m = 25\][/tex]
