Express \( \frac{15}{\sqrt{3}} - \sqrt{27} \) in the form \( k\sqrt{3} \), where \( k \) is an integer. - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 2
Question 4
Express \( \frac{15}{\sqrt{3}} - \sqrt{27} \) in the form \( k\sqrt{3} \), where \( k \) is an integer.
Worked Solution & Example Answer:Express \( \frac{15}{\sqrt{3}} - \sqrt{27} \) in the form \( k\sqrt{3} \), where \( k \) is an integer. - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 2
Step 1
Step 1: Simplify \( \sqrt{27} \)
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The square root of 27 can be simplified as follows:
[ \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} ]
Step 2
Step 2: Rewrite the expression
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Now that we have simplified ( \sqrt{27} ), we can rewrite the original expression:
[ \frac{15}{\sqrt{3}} - \sqrt{27} = \frac{15}{\sqrt{3}} - 3\sqrt{3} ]
Step 3
Step 3: Multiply numerator and denominator by \( \sqrt{3} \)
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To express ( \frac{15}{\sqrt{3}} ) in a more manageable form, we multiply the numerator and denominator by ( \sqrt{3} ):
[ \frac{15 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{15\sqrt{3}}{3} = 5\sqrt{3} ]
Step 4
Step 4: Combine the terms
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Now we substitute back into our expression:
[ 5\sqrt{3} - 3\sqrt{3} = (5 - 3)\sqrt{3} = 2\sqrt{3} ]
Thus, ( k = 2 ).
