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Chapter 0: Problem 52

Rationalize the denominator. $$\frac{5}{\sqrt{3}-1}$$

Short Answer

Expert verified
The rationized form of \( \frac{5}{\sqrt{3}-1} \) is \( \frac{5\sqrt{3}}{2} + \frac{5}{2} \).

Step by step solution

01

Multiply by the conjugate over itself

Multiply the given fraction by the conjugate of denominator over itself, \[\frac{5}{\sqrt{3}-1} * \frac{\sqrt{3}+1}{\sqrt{3}+1}\]
02

Perform the multiplication

First multiply the numerators together,\[5 * (\sqrt{3} + 1) = 5\sqrt{3} + 5\]Then multiply the denominators together, using the expansion formula \( (a+b) * (a-b) = a^{2} - b^{2} \),\[(\sqrt{3}+1) * (\sqrt{3}-1) = (\sqrt{3})^{2} - (1)^{2} = 3 - 1 = 2\]
03

Simplify the fraction

Divide the elements of the numerator by the denominator, \[\frac{5\sqrt{3}+5}{2} = \frac{5\sqrt{3}}{2} + \frac{5}{2}\]

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Most popular questions from this chapter

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. $$\text { a. } 3^{\frac{1}{2}} \square 3^{\frac{1}{3}}$$ $$\text { b. } \sqrt{7}+\sqrt{18} \square \sqrt{7}+18$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{4}-16\) is factored completely as \(\left(x^{2}+4\right)\left(x^{2}-4\right)\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?
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