Knowing how to simplify the square root of 80 is a useful skill because it helps you rewrite radical expressions in a form that is easier to calculate, compare, and use in more advanced algebra. Although 80 is not a perfect square, it contains a perfect-square factor that can be moved outside the radical. The result is (4\sqrt{5}), the simplest exact form of (\sqrt{80}).
Why Simplify Square Roots?
A square root is the inverse of squaring. If (x^2 = 80), then (x = \sqrt{80}). Because 80 is not a perfect square, its square root is irrational, meaning it cannot be expressed as a fraction of two integers The details matter here..
Even so, simplifying a radical does not mean making it a whole number. Instead, it means rewriting the expression in a shorter and more useful form. The simplified radical should have:
- An integer coefficient outside the radical
- No perfect-square factors left inside the radical
- A radicand that cannot be simplified further
As an example, (\sqrt{80}) becomes (4\sqrt{5}). This form is exact and is generally preferred over a long decimal approximation.
What Is the Simplest Form of (\sqrt{80})?
The simplest form is:
[ \boxed{4\sqrt{5}} ]
To understand why, remember the basic radical rule:
[ \sqrt{ab} = \sqrt{a}\times\sqrt{b} ]
This rule works when (a) and (b) are nonnegative numbers. It allows you to separate a product inside a radical into individual radicals. If one of the factors is a perfect square, its square root can then be written as a regular number Easy to understand, harder to ignore..
For (\sqrt{80}), find two factors whose product is 80 and where at least one factor is a perfect square:
[ 80 = 16 \times 5 ]
Both 16 and 5 are positive, so the rule applies:
[ \sqrt{80} = \sqrt{16\times5} ]
Next, simplify each radical:
[ \sqrt{16\times5} = \sqrt{16}\times\sqrt{5} ]
Since (\sqrt{16}=4), the expression becomes:
[ \sqrt{80}=4\sqrt{5} ]
The number 5 has no perfect-square factors other than 1, so the radical cannot be simplified any further.
Step-by-Step Method for Simplifying (\sqrt{80})
Step 1: Write 80 as a product
Choose a perfect-square factor and the remaining factor:
[ 80=16\times5 ]
Step 2: Apply the square-root product rule
[ \sqrt{80}=\sqrt{16\times5} ]
[ \sqrt{80}=\sqrt{16}\times\sqrt{5} ]
Step 3: Simplify the perfect square
[ \sqrt{16}=4 ]
Therefore:
[ \boxed{\sqrt{80}=4\sqrt{5}} ]
Step 4: Check the answer
You can verify the result by squaring (4\sqrt{5}):
[ (4\sqrt{5})^2=4^2\times(\sqrt{5})^2 ]
[ =16\times5=80 ]
Because the square of (4\sqrt{5}) is 80, the simplified expression is correct Worth knowing..
Using Prime Factorization
Prime factorization provides another reliable way to simplify the square root of 80. First, break 80 into prime factors:
[ 80=8\times10 ]
[ 80=2^3\times2\times5 ]
[ 80=2^4\times5 ]
The exponent 4 is even, so the factors of 2 can be paired:
[ 2^4=(2^2)^2 ]
Now take the square root:
[ \sqrt{80}=\sqrt{2^4\times5} ]
[ =\sqrt{2^4}\times\sqrt{5} ]
[ =2^2\sqrt{5} ]
[ =4\sqrt{5} ]
This method is especially useful when a number contains several repeated prime factors.
Trying Intermediate Forms
Sometimes the first factor pair you choose does not immediately
