Question
Question Express in simplest radioal form. \[ 8 \sqrt{40}+\sqrt{160} \]
Ask by Gibson Bowman. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \(20\sqrt{10}\).
Solution
Calculate or simplify the expression \( 8\sqrt{40}+\sqrt{160} \).
Calculate the value by following steps:
- step0: Calculate:
\(8\sqrt{40}+\sqrt{160}\)
- step1: Simplify the root:
\(8\times 2\sqrt{10}+\sqrt{160}\)
- step2: Simplify the root:
\(8\times 2\sqrt{10}+4\sqrt{10}\)
- step3: Multiply the terms:
\(16\sqrt{10}+4\sqrt{10}\)
- step4: Collect like terms:
\(\left(16+4\right)\sqrt{10}\)
- step5: Add the numbers:
\(20\sqrt{10}\)
The expression \(8\sqrt{40}+\sqrt{160}\) can be simplified to \(20\sqrt{10}\) in simplest radical form.
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Simplify this solution Extra Insights
To simplify \( 8\sqrt{40} + \sqrt{160} \), we can start by breaking down the square roots. The number \( 40 \) can be written as \( 4 \times 10 \), so: \[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \] Therefore: \[ 8\sqrt{40} = 8 \times 2\sqrt{10} = 16\sqrt{10} \] Now, simplify \( \sqrt{160} \). The number \( 160 \) can be expressed as \( 16 \times 10 \): \[ \sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10} \] Now we can combine the terms: \[ 8\sqrt{40} + \sqrt{160} = 16\sqrt{10} + 4\sqrt{10} = (16 + 4)\sqrt{10} = 20\sqrt{10} \] So, the expression in its simplest radical form is: \[ \boxed{20\sqrt{10}} \]
