CBC Kunming - No. 10 Senior High School
Kunming, Yunnan
12. Rationaliz the given radical expression. Show your complete work.
Canama Datisis Columbia
a. $\frac{-\sqrt{5}}{\sqrt{7}-3}$
b. $\frac{5 \sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
[2]
13. Determine whether the given value of x is a root of the equation.
[2]
a. $\sqrt{2 x+1}=9 x ; x=16$
b. $\sqrt{3 x-6}=6 ; x=14$
14. Which of these equations have real roots? Justify your answers.
[4]
a. $2 \sqrt{x+B}=3 \sqrt{3 x+1}$
b. $2 \sqrt{x-8}=3 \sqrt{3 x+1}$
15. Solve the value of $x$ and check if there is extraneous roots.
[3]
$$
\sqrt{x+3}-=\sqrt{3 x-2}
$$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To rationalize the expression $\frac{-\sqrt{5}}{\sqrt{7}-3}$, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{7}+3$. ‖ ‖ step 2 ⋮ This gives us $\frac{-\sqrt{5} (\sqrt{7}+3)}{(\sqrt{7}-3)(\sqrt{7}+3)}$. The denominator simplifies to $7 - 9 = -2$. ‖ ‖ step 3 ⋮ Therefore, we have $\frac{-\sqrt{5} (\sqrt{7}+3)}{-2} = \frac{\sqrt{5} (\sqrt{7}+3)}{2}$. ‖ ‖ step 4 ⋮ This can be further simplified to $\frac{3\sqrt{5}}{2} + \frac{\sqrt{35}}{2}$. ‖ ‖ step 5 ⋮ The final result is $\frac{3\sqrt{5} + \sqrt{35}}{2}$. ‖ ‖ step 6 ⋮ Now, for the second expression $\frac{5\sqrt{3} + \sqrt{2}}{\sqrt{3}-\sqrt{2}}$, we again multiply by the conjugate $\sqrt{3} + \sqrt{2}$. ‖ ‖ step 7 ⋮ This results in $\frac{(5\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}$. The denominator simplifies to $3 - 2 = 1$. ‖ ‖ step 8 ⋮ The numerator expands to $5\cdot3 + 5\sqrt{6} + \sqrt{6} + 2 = 17 + 6\sqrt{6}$. ‖ ‖ step 9 ⋮ Thus, the final result is $17 + 6\sqrt{6}$. ‖ ∻Answer∻ ⚹ The rationalized forms are $\frac{3\sqrt{5} + \sqrt{35}}{2}$ and $17 + 6\sqrt{6}$. ⚹ ∻Key Concept∻ ⚹ Rationalizing expressions involves eliminating radicals from the denominator by multiplying by the conjugate. ⚹ ∻Explanation∻ ⚹ The process of rationalization simplifies the expressions, making them easier to work with in further calculations. ⚹◊What method can you use to rationalize the denominator of the expression $ \frac{-\sqrt{5}}{\sqrt{7}-3} $?⍭ Generate me a similar question◊

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