A5 Find the value of $$ \sqrt{\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}} $$, giving your answer in the form $$ a+b \sqrt{2} $$, where $$ a $$ and $$ b $$ are integers.

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UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \sqrt{\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{3+2\sqrt{2}}{3-2\sqrt{2}}}$$
- step1: Complete the square:
  $$\sqrt{\left(\frac{\sqrt{2}+1}{\sqrt{2}-1}ight)^{2}}$$
- step2: Simplify the root:
  $$\frac{\sqrt{2}+1}{\sqrt{2}-1}$$
- step3: Multiply by the Conjugate:
  $$\frac{\left(\sqrt{2}+1ight)\left(\sqrt{2}+1ight)}{\left(\sqrt{2}-1ight)\left(\sqrt{2}+1ight)}$$
- step4: Multiply the numbers:
  $$\frac{\left(\sqrt{2}+1ight)\left(\sqrt{2}+1ight)}{1}$$
- step5: Divide the numbers:
  $$\left(\sqrt{2}+1ight)\left(\sqrt{2}+1ight)$$
- step6: Apply the distributive property:
  $$\sqrt{2}	imes \sqrt{2}+\sqrt{2}	imes 1+1	imes \sqrt{2}+1	imes 1$$
- step7: Calculate the product:
  $$2+\sqrt{2}+\sqrt{2}+1$$
- step8: Add the numbers:
  $$3+2\sqrt{2}$$
The value of $$ \sqrt{\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}} $$ is $$ 3+2\sqrt{2} $$.
The value is $$ 3 + 2\sqrt{2} $$.

A5 Find the value of , giving
your answer in the form ,
where and are integers.

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A5 Find the value of , giving your answer in the form , where and are integers.