Combine these radicals. square root of 27 -square root of 3
The correct answer and explanation is:
Answer: 27−3=23\sqrt{27} – \sqrt{3} = \boxed{2\sqrt{3}}
Explanation (300 words):
To simplify and combine the radicals 27−3\sqrt{27} – \sqrt{3}, we begin by simplifying each square root.
Step 1: Simplify 27\sqrt{27}
The number 27 is not a perfect square, but it has a factor that is: 27=9×327 = 9 \times 3
Now, apply the property of square roots: 27=9×3=9⋅3\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3}
Since 9=3\sqrt{9} = 3, this becomes: 27=33\sqrt{27} = 3\sqrt{3}
Step 2: Rewrite the Expression
Now that we’ve simplified 27\sqrt{27}, substitute it back into the original expression: 27−3=33−3\sqrt{27} – \sqrt{3} = 3\sqrt{3} – \sqrt{3}
Step 3: Combine Like Terms
Both terms involve the same radical: 3\sqrt{3}. Just like combining like terms in algebra (e.g., 3x−x=2x3x – x = 2x), you can combine radical terms with the same radicand (the number inside the square root). 33−3=(3−1)3=233\sqrt{3} – \sqrt{3} = (3 – 1)\sqrt{3} = 2\sqrt{3}
So, the simplified expression is: 23\boxed{2\sqrt{3}}
Why This Works:
The process of simplifying radicals involves factoring out perfect squares. Once all terms are expressed with the same radical part, you treat them like variables and combine coefficients. In this case, both terms had 3\sqrt{3}, allowing us to combine them easily. This is a fundamental algebraic skill and is important in higher math where radicals appear in equations and expressions that must be simplified before solving.