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Simplify $$ oot{32} + oot{18}$$ giving your answer in the form $a\sqrt{2}$, where $a$ is an integer - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 1

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Simplify--$$-oot{32}-+--oot{18}$$--giving-your-answer-in-the-form-$a\sqrt{2}$,-where-$a$-is-an-integer-Edexcel-A-Level Maths Pure-Question 4-2012-Paper 1.png

Simplify $$ oot{32} + oot{18}$$ giving your answer in the form $a\sqrt{2}$, where $a$ is an integer. --- Simplify $$\frac{\root{32} + \root{18}}{3 + \sqrt{2}}$... show full transcript

Worked Solution & Example Answer:Simplify $$ oot{32} + oot{18}$$ giving your answer in the form $a\sqrt{2}$, where $a$ is an integer - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 1

Step 1

Simplify $$\root{32} + \root{18}$$

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Answer

To simplify the expression, we first need to break down the square roots:

  1. Calculating each square root:

    • For \root32\root{32}: \root32=\root16⋅2=\root16⋅\root2=42\root{32} = \root{16 \cdot 2} = \root{16} \cdot \root{2} = 4\sqrt{2}
    • For \root18\root{18}: \root18=\root9⋅2=\root9⋅\root2=32\root{18} = \root{9 \cdot 2} = \root{9} \cdot \root{2} = 3\sqrt{2}

  2. Combining the results: \root32+\root18=42+32=(4+3)2=72\root{32} + \root{18} = 4\sqrt{2} + 3\sqrt{2} = (4 + 3)\sqrt{2} = 7\sqrt{2}

Thus, the final answer is:

727\sqrt{2}

Step 2

Simplify $$\frac{\root{32} + \root{18}}{3 + \sqrt{2}}$$

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Answer

To simplify this expression, we can follow these steps:

  1. Use the result from part (a): We know that the numerator is: \root32+\root18=72\root{32} + \root{18} = 7\sqrt{2} Hence, we can rewrite our expression as: 723+2\frac{7\sqrt{2}}{3 + \sqrt{2}}

  2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 3−23 - \sqrt{2}: 72(3−2)(3+2)(3−2)\frac{7\sqrt{2} (3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})}

  3. Calculate the denominator: (3+2)(3−2)=32−(2)2=9−2=7 (3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7

  4. Calculate the numerator: 72(3−2)=212−77\sqrt{2} (3 - \sqrt{2}) = 21\sqrt{2} - 7

  5. Final form: Now the expression is: 212−77=2172−77=32−1\frac{21\sqrt{2} - 7}{7} = \frac{21}{7}\sqrt{2} - \frac{7}{7} = 3\sqrt{2} - 1

Thus, in the form b2+cb\sqrt{2} + c where b=3b = 3 and c=−1c = -1, the final answer is: 32−13\sqrt{2} - 1

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