Question
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Answer
\( a=61 \)
Solution
Solve the equation \( \sqrt{a+3}=8 \).
Solve the equation by following steps:
- step0: Solve for \(a\):
\(\sqrt{a+3}=8\)
- step1: Find the domain:
\(\sqrt{a+3}=8,a\geq -3\)
- step2: Raise both sides to the \(2\)-th power\(:\)
\(\left(\sqrt{a+3}\right)^{2}=8^{2}\)
- step3: Evaluate the power:
\(a+3=64\)
- step4: Move the constant to the right side:
\(a=64-3\)
- step5: Subtract the numbers:
\(a=61\)
- step6: Check if the solution is in the defined range:
\(a=61,a\geq -3\)
- step7: Find the intersection:
\(a=61\)
- step8: Check the solution:
\(a=61\)
The solution to the equation \( \sqrt{a+3}=8 \) is \( a=61 \).
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Simplify this solution Bonus Knowledge
To solve for \( a \), we first eliminate the square root by squaring both sides of the equation. Thus, we obtain \( a + 3 = 8^2 \), which simplifies to \( a + 3 = 64 \). Next, we isolate \( a \) by subtracting 3 from both sides, resulting in \( a = 64 - 3 \), or \( a = 61 \). A fun way to verify your solution is to plug \( a \) back into the original equation: \( \sqrt{61 + 3} = \sqrt{64} = 8 \). Huzzah! Your answer checks out!
