19 (a) Show that $\sqrt{11} \times \sqrt{22} = 11 \sqrt{2}$ - OCR - GCSE Maths - Question 19 - 2023 - Paper 6
Question 19
19 (a) Show that $\sqrt{11} \times \sqrt{22} = 11 \sqrt{2}$.
(b) Show that $\frac{\sqrt{11}}{13 + \sqrt{22}}$ can be written in the form $\frac{a \sqrt{11} - 11 \s... show full transcript
Worked Solution & Example Answer:19 (a) Show that $\sqrt{11} \times \sqrt{22} = 11 \sqrt{2}$ - OCR - GCSE Maths - Question 19 - 2023 - Paper 6
Step 1
Show that $\sqrt{11} \times \sqrt{22} = 11 \sqrt{2}$
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Answer
To prove that 11×22=112, we can use the property of square roots which states that a×b=ab.
Hence, we have:
11×22=11×22
Calculating 11×22 gives:
11×22=242
Next, we can simplify 242. Since 242=11×22=11×(11×2), we can write:
242=112×2=112
Thus, we have shown that 11×22=112.
Step 2
Show that $\frac{\sqrt{11}}{13 + \sqrt{22}}$ can be written in the form $\frac{a \sqrt{11} - 11 \sqrt{2}}{b}$ where $a$ and $b$ are integers.
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Answer
To rewrite 13+2211, we will multiply the numerator and the denominator by the conjugate of the denominator, which is 13−22.
Doing so gives:
(13+22)(13−22)11(13−22)
Next, we can calculate the denominator:
(13+22)(13−22)=132−(22)2=169−22=147
Now, calculating the numerator yields:
11(13−22)=1311−11×22=1311−242
Since 242=112, substituting gives:
=1311−112
Therefore, we have:
13+2211=1471311−112
In this form, it is clear that a=13 and b=147, hence proving that it can be expressed in the desired format.
