Chapter 12

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are real numbers, then \((a+b)^{2}=a^{2}+b^{2}\)

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ a=3, c=4 $$

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (5,8),(-2,3) $$

Solve the quadratic equation by completing the square. $$ x^{2}+14 x=-13 $$

Find the midpoint of the line segment with the given endpoints. Then show that the midpoint is the same distance from each given point. \((-3,6),(1,8)\)

Rewrite the expression using rational exponent notation. $$ (\sqrt{16})^{5} $$

Simplify the expression. $$ 5 \sqrt{7}+2 \sqrt{7} $$

Solve the equation. Check for extraneous solutions. $$ -5+\sqrt{x}=0 $$

Find the domain and the range of the function. $$y=\sqrt{x+5}$$

Find a counterexample to show that the statement is not true. If \(a, b,\) and \(c\) are nonzero real numbers, then \((a \div b) \div c=a \div(b \div c)\) (Note: The counterexample shows that the associative property does not hold for division.)

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-3,1),(2,6) $$

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ a=10, b=24 $$

Find the midpoint of the line segment with the given endpoints. Then show that the midpoint is the same distance from each given point. \((0,0),(-8,12)\)

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}-x-2=0 $$

Rewrite the expression using radical notation. $$ 6^{1 / 3} $$

Simplify the expression. $$ \sqrt{3}+5 \sqrt{3} $$

Solve the equation. Check for extraneous solutions. $$ x=\sqrt{5 x+24} $$

Find the domain and the range of the function. $$y=\sqrt{x-10}$$

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are integers, then \(a \div b\) is an integer.

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-6,-2),(-3,-5) $$

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ b=3, c=7 $$

Find the midpoint of the line segment connecting the given points. \((1,2),(5,4)\)

Choose a method and solve the quadratic equation. Explain your choice. $$ 3 x^{2}+17 x+10=0 $$

Rewrite the expression using radical notation. $$ 7^{1 / 2} $$

Simplify the expression. $$ 11 \sqrt{3}-12 \sqrt{3} $$

Solve the equation. Check for extraneous solutions. $$ \sqrt{5 x+1}+8=12 $$

Find the domain of the function. Then sketch its graph. $$y=4 \sqrt{x}$$

Find a counterexample to show that the statement is not true. If \(a>4,\) then \(\sqrt{a}\) is not rational.

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (4,5),(-1,3) $$

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ b=9, c=16 $$

Find the midpoint of the line segment connecting the given points. \((0,0),(0,8)\)

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}-9=0 $$

Rewrite the expression using radical notation. $$ 10^{3 / 2} $$

Simplify the expression. $$ 2 \sqrt{6}-\sqrt{6} $$

Solve the equation. Check for extraneous solutions. $$ \sqrt{4 x+5}=x $$

Find the domain of the function. Then sketch its graph. $$y=\sqrt{x}+5$$

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ a=5, c=10 $$

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-6,1),(3,1) $$

Find the midpoint of the line segment connecting the given points. \((-1,2),(7,4)\)

Choose a method and solve the quadratic equation. Explain your choice. $$ -3 x^{2}+5 x+5=0 $$

Rewrite the expression using radical notation. $$ 8^{7 / 3} $$

Simplify the expression. $$ 4 \sqrt{5}+\sqrt{3}+\sqrt{5} $$

Solve the equation. Check for extraneous solutions. $$ \sqrt{x+6}=x $$

Find the domain of the function. Then sketch its graph. $$y=3 \sqrt{x+1}$$

USING THE PYTHAGOREAN THEOREM Find the missing length of the right triangle if \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse. $$ a=14, c=21 $$

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-2,-1),(3,-3) $$

Find the midpoint of the line segment connecting the given points. \((0,-3),(-4,2)\)

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}+2 x-14=0 $$

Evaluate the expression without using a calculator. $$ \sqrt[3]{8} $$

Simplify the expression. $$ 3 \sqrt{11}-\sqrt{5}+\sqrt{11} $$