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Write \( \sqrt{75} - \sqrt{27} \) in the form \( k \sqrt{x} \), where \( k \) and \( x \) are integers. - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 1

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Write--\(-\sqrt{75}---\sqrt{27}-\)-in-the-form-\(-k-\sqrt{x}-\),-where-\(-k-\)-and-\(-x-\)-are-integers.-Edexcel-A-Level Maths Pure-Question 3-2010-Paper 1.png

Write \( \sqrt{75} - \sqrt{27} \) in the form \( k \sqrt{x} \), where \( k \) and \( x \) are integers.

Worked Solution & Example Answer:Write \( \sqrt{75} - \sqrt{27} \) in the form \( k \sqrt{x} \), where \( k \) and \( x \) are integers. - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 1

Step 1

Step 1: Simplifying \( \sqrt{75} \)

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Answer

To simplify ( \sqrt{75} ), we can factor it as:

75=25⋅3=25⋅3=53\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}

Step 2

Step 2: Simplifying \( \sqrt{27} \)

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Answer

Similarly, we simplify ( \sqrt{27} ):

27=9⋅3=9⋅3=33\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}

Step 3

Step 3: Combining the terms

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Answer

Now we can substitute these results into our original expression:

75−27=53−33=(5−3)3=23\sqrt{75} - \sqrt{27} = 5\sqrt{3} - 3\sqrt{3} = (5 - 3)\sqrt{3} = 2\sqrt{3}

This gives us the final result in the required form ( k\sqrt{x} ) where ( k = 2 ) and ( x = 3 ).

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