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Section 7
Rationalizing a Denominator
Justin simplified $\frac{7}{2 \sqrt{7}}$ by first writing 7 as $\sqrt{49}$ and then dividing numerator and denominator by $\sqrt{7}$.a. Show that Justin's solution is correct.b. Can $\frac{7}{2 \sqrt{5}}$ be simplified by using the same procedure? Explain why or why not.
To rationalize the denominator of $\frac{4}{2+\sqrt{8}},$ Brittany multiplied by $\frac{2-\sqrt{8}}{2}$ and Justin multiplied by $\frac{1-\sqrt{2}}{1-\sqrt{2}} .$ Explain why both are correct.
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{\sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{5}{\sqrt{10}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{\sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{2 \sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{15}{5 \sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{8 \sqrt{6}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{12}{\sqrt{27}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{6}{\sqrt{12}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{8}{\sqrt{24}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{2 \sqrt{2}}{4 \sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{5 \sqrt{5}}{15 \sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{24}}{2 \sqrt{6}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{3+\sqrt{5}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{5-\sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{1+\sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{3-\sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{3}{1+\sqrt{5}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{4+\sqrt{7}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{3}{\sqrt{5}-2}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{9}{\sqrt{7}+2}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{2}}{2-\sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{6}{3+\sqrt{3}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{5 x}}{\sqrt{5 x}-2}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{20 y}}{y \sqrt{5}+1}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{2+\sqrt{2}}{3-\sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{6-\sqrt{7}}{4-\sqrt{7}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{10}-1}{\sqrt{10}+1}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{7+\sqrt{5}}{7-\sqrt{5}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{a+2}{b-\sqrt{2}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{2 \sqrt{x}-5 \sqrt{y}}{\sqrt{x}+\sqrt{y}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{4}{\sqrt{z}+8}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{\sqrt{a}}{\sqrt{a}-2}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{2}{\sqrt{x}}+\frac{2}{\sqrt{y}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{\sqrt{x}+6}-\frac{2}{\sqrt{6}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{2 \sqrt{a}}{\sqrt{b}}+\frac{2 \sqrt{b}}{\sqrt{a}}$
Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{3}{x+\sqrt{2}}+\frac{5}{\sqrt{x}}$
a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form.$\frac{4}{\sqrt{6}}$
a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form.$\frac{2+\sqrt{3}}{\sqrt{3}}$
a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form.$\frac{4}{\sqrt{3}-1}$
a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form.$\frac{3+\sqrt{7}}{3-\sqrt{7}}$
Solve and check each equation.$2 a+\sqrt{50}=\sqrt{98}$
Solve and check each equation.$5 x-\sqrt{12}=\sqrt{108}-3 x$
Solve and check each equation.$y \sqrt{3}+1=3-y$
Solve and check each equation.$7-b \sqrt{8}=b \sqrt{5}+4$
The area of a rectangle is 24 square inches. The length of the rectangle is $\sqrt{5}+1$ inches. Express the width of the rectangle in simplest form.
The perimeter of an isosceles triangle is $\sqrt{50}$ feet. The lengths of the sides are in the ratio $3 : 3 : 4$ . Find the length of each side of the triangle.