Rationalise the denominators of 1/√2 + √3 + √5

Multiply by Conjugate: To rationalize the denominator of the expression 1/(√2 + √3 + √5), we need to multiply the numerator and the denominator by the conjugate of the denominator. However, in this case, the denominator is not a simple binomial, but a sum of three square roots. We will need to rationalize step by step, starting with two terms at a time.Deal with √2 and √3: First, let's rationalize the denominator by dealing with √2 and √3. We will multiply the numerator and the denominator by the conjugate of (√2 + √3), which is (√2 - √3). (1/(√2 + √3 + √5)) * ((√2 - √3)/(√2 - √3))Deal with -1 and √10: Now, let's perform the multiplication in the denominator: (√2 + √3 + √5) * (√2 - √3) = (√2 * √2) - (√3 * √2) + (√5 * √2) + (√2 * √3) - (√3 * √3) + (√5 * √3) This simplifies to: 2 - √6 + √10 + √6 - 3 + √15 Which further simplifies to: -1 + √10 + √15Deal with -9 and √150: Now we have a new expression: (1 * (√2 - √3)) / (-1 + √10 + √15) This simplifies to: (√2 - √3) / (-1 + √10 + √15)Next, we need to rationalize the denominator again, this time dealing with -1 and √10. We will multiply the numerator and the denominator by the conjugate of (-1 + √10), which is (-1 - √10). ((√2 - √3) / (-1 + √10 + √15)) * ((-1 - √10) / (-1 - √10))Perform the multiplication in the denominator: (-1 + √10 + √15) * (-1 - √10) = (-1 * -1) + (√10 * -1) + (√15 * -1) - (√10 * -1) - (√10 * √10) - (√15 * √10) This simplifies to: 1 - √10 - √15 + √10 - 10 - √150 Which further simplifies to: -9 - √150Now we have a new expression: ((√2 - √3) * (-1 - √10)) / (-9 - √150) This simplifies to: (-√2 - √20 + √3 + √30) / (-9 - √150)Finally, we need to rationalize the denominator one last time, dealing with -9 and √150. We will multiply the numerator and the denominator by the conjugate of (-9 + √150), which is (-9 - √150). ((-√2 - √20 + √3 + √30) / (-9 - √150)) * ((-9 + √150) / (-9 + √150))Perform the multiplication in the denominator: (-9 - √150) * (-9 + √150) = (-9 * -9) + (√150 * 9) - (√150 * 9) + (√150 * √150) This simplifies to: 81 - 150 Which further simplifies to: -69Now we have a new expression: ((-√2 - √20 + √3 + √30) * (-9 + √150)) / -69 This simplifies to: (9√2 + 9√20 - 9√3 - 9√30 - √300 - √3000 + √450 + √4500) / -69Simplify the numerator by combining like terms and then divide each term by -69: (9√2/ -69) + (9√20/ -69) - (9√3/ -69) - (9√30/ -69) - (√300/ -69) - (√3000/ -69) + (√450/ -69) + (√4500/ -69) This simplifies to: (-√2/ -7) + (-√20/ -7) + (√3/ -7) + (√30/ -7) + (√300/ -69) + (√3000/ -69) - (√450/ -69) - (√4500/ -69)

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Rationalise the denominators of $\frac{1}{\sqrt{2}} + \sqrt{3} + \sqrt{5}$

View step-by-step help

Home

Math Problems

Algebra 2

Simplify radical expressions using conjugates

Full solution

Q. Rationalise the denominators of

\frac{1}{\sqrt{2}} + \sqrt{3} + \sqrt{5}

Multiply by Conjugate: To rationalize the denominator of the expression $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ , we need to multiply the numerator and the denominator by the conjugate of the denominator. However, in this case, the denominator is not a simple binomial, but a sum of three square roots. We will need to rationalize step by step, starting with two terms at a time.
Deal with $\sqrt{2}$ and $\sqrt{3}$ : First, let's rationalize the denominator by dealing with $\sqrt{2}$ and $\sqrt{3}$ . We will multiply the numerator and the denominator by the conjugate of $(\sqrt{2} + \sqrt{3})$ , which is $(\sqrt{2} - \sqrt{3})$ . $\newline$ $\frac{1}{(\sqrt{2} + \sqrt{3} + \sqrt{5})} \times \frac{(\sqrt{2} - \sqrt{3})}{(\sqrt{2} - \sqrt{3})}$
Deal with $-1$ and $\sqrt{10}$ : Now, let's perform the multiplication in the denominator: $\newline$ $(\sqrt{2} + \sqrt{3} + \sqrt{5}) \times (\sqrt{2} - \sqrt{3}) = (\sqrt{2} \times \sqrt{2}) - (\sqrt{3} \times \sqrt{2}) + (\sqrt{5} \times \sqrt{2}) + (\sqrt{2} \times \sqrt{3}) - (\sqrt{3} \times \sqrt{3}) + (\sqrt{5} \times \sqrt{3})$ $\newline$ This simplifies to: $\newline$ $2 - \sqrt{6} + \sqrt{10} + \sqrt{6} - 3 + \sqrt{15}$ $\newline$ Which further simplifies to: $\newline$ $-1 + \sqrt{10} + \sqrt{15}$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $\sqrt{150}$ $0$ $\newline$ Which further simplifies to: $\newline$ $\sqrt{150}$ $1$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $1 - \sqrt{10} - \sqrt{15} + \sqrt{10} - 10 - \sqrt{150}$ $\newline$ Which further simplifies to: $\newline$ $-9 - \sqrt{150}$ Now we have a new expression: $\newline$ $((\sqrt{2} - \sqrt{3}) \times (-1 - \sqrt{10})) / (-9 - \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $1 - \sqrt{10} - \sqrt{15} + \sqrt{10} - 10 - \sqrt{150}$ $\newline$ Which further simplifies to: $\newline$ $-9 - \sqrt{150}$ Now we have a new expression: $\newline$ $((\sqrt{2} - \sqrt{3}) \times (-1 - \sqrt{10})) / (-9 - \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})$ Finally, we need to rationalize the denominator one last time, dealing with $-9$ and $\sqrt{150}$ . We will multiply the numerator and the denominator by the conjugate of $(-9 + \sqrt{150})$ , which is $(-9 - \sqrt{150})$ . $\newline$ $((-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})) \times ((-9 + \sqrt{150}) / (-9 + \sqrt{150}))$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $1 - \sqrt{10} - \sqrt{15} + \sqrt{10} - 10 - \sqrt{150}$ $\newline$ Which further simplifies to: $\newline$ $-9 - \sqrt{150}$ Now we have a new expression: $\newline$ $((\sqrt{2} - \sqrt{3}) \times (-1 - \sqrt{10})) / (-9 - \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})$ Finally, we need to rationalize the denominator one last time, dealing with $-9$ and $\sqrt{150}$ . We will multiply the numerator and the denominator by the conjugate of $(-9 + \sqrt{150})$ , which is $(-9 - \sqrt{150})$ . $\newline$ $((-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})) \times ((-9 + \sqrt{150}) / (-9 + \sqrt{150}))$ Perform the multiplication in the denominator: $\newline$ $(-9 - \sqrt{150}) \times (-9 + \sqrt{150}) = (-9 \times -9) + (\sqrt{150} \times 9) - (\sqrt{150} \times 9) + (\sqrt{150} \times \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $0$ $\newline$ Which further simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $1$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $1 - \sqrt{10} - \sqrt{15} + \sqrt{10} - 10 - \sqrt{150}$ $\newline$ Which further simplifies to: $\newline$ $-9 - \sqrt{150}$ Now we have a new expression: $\newline$ $((\sqrt{2} - \sqrt{3}) \times (-1 - \sqrt{10})) / (-9 - \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})$ Finally, we need to rationalize the denominator one last time, dealing with $-9$ and $\sqrt{150}$ . We will multiply the numerator and the denominator by the conjugate of $(-9 + \sqrt{150})$ , which is $(-9 - \sqrt{150})$ . $\newline$ $((-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})) \times ((-9 + \sqrt{150}) / (-9 + \sqrt{150}))$ Perform the multiplication in the denominator: $\newline$ $(-9 - \sqrt{150}) \times (-9 + \sqrt{150}) = (-9 \times -9) + (\sqrt{150} \times 9) - (\sqrt{150} \times 9) + (\sqrt{150} \times \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $0$ $\newline$ Which further simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $1$ Now we have a new expression: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $2$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $3$
Deal with $-9$ and $\sqrt{150}$ : Now we have a new expression: $\newline$ $(1 \times (\sqrt{2} - \sqrt{3})) / (-1 + \sqrt{10} + \sqrt{15})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ Next, we need to rationalize the denominator again, this time dealing with $-1$ and $\sqrt{10}$ . We will multiply the numerator and the denominator by the conjugate of $(-1 + \sqrt{10})$ , which is $(-1 - \sqrt{10})$ . $\newline$ $((\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})) \times ((-1 - \sqrt{10}) / (-1 - \sqrt{10}))$ Perform the multiplication in the denominator: $\newline$ $(-1 + \sqrt{10} + \sqrt{15}) \times (-1 - \sqrt{10}) = (-1 \times -1) + (\sqrt{10} \times -1) + (\sqrt{15} \times -1) - (\sqrt{10} \times -1) - (\sqrt{10} \times \sqrt{10}) - (\sqrt{15} \times \sqrt{10})$ $\newline$ This simplifies to: $\newline$ $1 - \sqrt{10} - \sqrt{15} + \sqrt{10} - 10 - \sqrt{150}$ $\newline$ Which further simplifies to: $\newline$ $-9 - \sqrt{150}$ Now we have a new expression: $\newline$ $((\sqrt{2} - \sqrt{3}) \times (-1 - \sqrt{10})) / (-9 - \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})$ Finally, we need to rationalize the denominator one last time, dealing with $-9$ and $\sqrt{150}$ . We will multiply the numerator and the denominator by the conjugate of $(-9 + \sqrt{150})$ , which is $(-9 - \sqrt{150})$ . $\newline$ $((-\sqrt{2} - \sqrt{20} + \sqrt{3} + \sqrt{30}) / (-9 - \sqrt{150})) \times ((-9 + \sqrt{150}) / (-9 + \sqrt{150}))$ Perform the multiplication in the denominator: $\newline$ $(-9 - \sqrt{150}) \times (-9 + \sqrt{150}) = (-9 \times -9) + (\sqrt{150} \times 9) - (\sqrt{150} \times 9) + (\sqrt{150} \times \sqrt{150})$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $0$ $\newline$ Which further simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $1$ Now we have a new expression: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $2$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $3$ Simplify the numerator by combining like terms and then divide each term by $\sqrt{150}$ $0$ : $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $4$ $\newline$ This simplifies to: $\newline$ $(\sqrt{2} - \sqrt{3}) / (-1 + \sqrt{10} + \sqrt{15})$ $5$

Rationalise the denominators of $\frac{1}{\sqrt{2}} + \sqrt{3} + \sqrt{5}$

Full solution

More problems from Simplify radical expressions using conjugates

Related topics

Rationalise the denominators of 12+3+5\frac{1}{\sqrt{2}} + \sqrt{3} + \sqrt{5}2​1​+3​+5​

Rationalise the denominators of $\frac{1}{\sqrt{2}} + \sqrt{3} + \sqrt{5}$