Problem 98
Question
Simplify. $$ \frac{8-\sqrt{80}}{4} $$
Step-by-Step Solution
Verified Answer
2 - \(\root{5}\)
1Step 1: Simplify the Square Root
Simplify the term inside the square root. Notice that \(\root{80}\) can be expressed as \(\root{16 \times 5}\). Then, we have \(\root{80} = \root{16 \times 5} = \root{16} \times \root{5} = 4\root{5}\).
2Step 2: Substitute Simplified Square Root Back
Substitute \(4\root{5}\) back into the expression to get: \(\frac{8 - 4\root{5}}{4}\).
3Step 3: Split the Fraction
Separate the fraction into two parts: \(\frac{8}{4} - \frac{4\root{5}}{4}\).
4Step 4: Simplify Each Term
Simplify each term separately: \(\frac{8}{4} = 2\) and \(\frac{4\root{5}}{4} = \root{5}\). Therefore, the expression is \2 - \root{5}\.
Key Concepts
Simplifying Square RootsFraction OperationsAlgebraic Simplification
Simplifying Square Roots
Understanding how to simplify square roots makes solving algebraic expressions easier. Start by looking for a factor of the number inside the square root that is a perfect square. For example, in \(\root{80}\), 80 can be broken down into 16 and 5, because 16 is a perfect square.
So, \(\sqrt{80}\ = \sqrt{16 \times 5}\ = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\).
Substituting these back into the expression makes further simplification possible. Leading to a cleaner and easier-to-solve expression, simplifying square roots like this is foundational for handling higher algebra problems.
So, \(\sqrt{80}\ = \sqrt{16 \times 5}\ = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}\).
Substituting these back into the expression makes further simplification possible. Leading to a cleaner and easier-to-solve expression, simplifying square roots like this is foundational for handling higher algebra problems.
Fraction Operations
When faced with complex fractions, breaking them down step by step is essential. After simplifying square roots, let's tackle the fraction \(\frac{8 - 4\sqrt{5}}{4}\).
Begin by splitting the fraction: \(\frac{8}{4} - \frac{4\sqrt{5}}{4}\).
This separation allows you to simplify each term individually. By doing so, \(\frac{8}{4} = 2\) and \(\frac{4\sqrt{5}}{4} = \sqrt{5}\).
Understanding fraction operations and being comfortable with these algebraic steps is crucial for simplifying complex algebraic expressions.
Begin by splitting the fraction: \(\frac{8}{4} - \frac{4\sqrt{5}}{4}\).
This separation allows you to simplify each term individually. By doing so, \(\frac{8}{4} = 2\) and \(\frac{4\sqrt{5}}{4} = \sqrt{5}\).
Understanding fraction operations and being comfortable with these algebraic steps is crucial for simplifying complex algebraic expressions.
Algebraic Simplification
Algebraic simplification brings order to complex expressions. By simplifying components such as square roots and fractions, the overall algebraic expression becomes more manageable.
For example, after simplifying and substituting the square root back into the expression, we ended up with \(\frac{8 - 4\sqrt{5}}{4}\). We then split this into two simpler fractions: \(\frac{8}{4}\), which simplifies to 2, and \(\frac{4\sqrt{5}}{4}\) which simplifies to \(\root{5}\).
The final simplified expression is \(2 - \sqrt{5}\).
Through careful steps focusing on square roots and fractions, algebraic simplification brings clarity and paves the way for solving more advanced problems.
For example, after simplifying and substituting the square root back into the expression, we ended up with \(\frac{8 - 4\sqrt{5}}{4}\). We then split this into two simpler fractions: \(\frac{8}{4}\), which simplifies to 2, and \(\frac{4\sqrt{5}}{4}\) which simplifies to \(\root{5}\).
The final simplified expression is \(2 - \sqrt{5}\).
Through careful steps focusing on square roots and fractions, algebraic simplification brings clarity and paves the way for solving more advanced problems.