What is the exact value of the expression square root of 486 – square root of 24 + square root of 6?
The correct answer and explanation is:
To find the exact value of the expression 486−24+6\sqrt{486} – \sqrt{24} + \sqrt{6}, let’s break it down into manageable steps:
Step 1: Simplify each square root
- 486\sqrt{486}: Start by factoring 486. 486=2×35486 = 2 \times 3^5 So, 486=2×35=2×34×3=32×2×3=96\sqrt{486} = \sqrt{2 \times 3^5} = \sqrt{2} \times \sqrt{3^4 \times 3} = 3^2 \times \sqrt{2 \times 3} = 9 \sqrt{6}
- 24\sqrt{24}: Factor 24 as: 24=23×324 = 2^3 \times 3 So, 24=23×3=26\sqrt{24} = \sqrt{2^3 \times 3} = 2 \sqrt{6}
- 6\sqrt{6} is already in its simplest form.
Step 2: Substitute the simplified square roots
Now, substitute the simplified square roots back into the expression: 486−24+6=96−26+6\sqrt{486} – \sqrt{24} + \sqrt{6} = 9\sqrt{6} – 2\sqrt{6} + \sqrt{6}
Step 3: Combine like terms
Since all the terms involve 6\sqrt{6}, we can factor out 6\sqrt{6}: (9−2+1)6=86(9 – 2 + 1)\sqrt{6} = 8\sqrt{6}
Final Answer:
The exact value of the expression is 868\sqrt{6}.
Explanation:
The process involves simplifying each square root by factoring the numbers into their prime components and then extracting perfect squares. Once the square roots are simplified, you can combine like terms since all three terms involve 6\sqrt{6}. The result is 868\sqrt{6}, which is the exact simplified form of the expression.