Chapter 10

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ g(0) $$

Use rational expressions to write as a single radical expression. $$ \sqrt[4]{5} \cdot \sqrt[3]{x} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. \((-\sqrt{5}, 0)\) and \((0, \sqrt{7})\)

Factor each numerator and denominator. Then simplify if possible. $$ \frac{-5+10 \sqrt{7}}{5} $$

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. \(\frac{9}{\sqrt[3]{5}}\)

Consider the equations \(\sqrt{2 x}=4\) and \(\sqrt[3]{2 x}=4\) a. Explain the difference in solving these equations. b. Explain the similarity in solving these equations.

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ g(7) $$

Use rational expressions to write as a single radical expression. $$ \sqrt{5 r} \cdot \sqrt[3]{s} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (1.7,-3.6) and (-8.6,5.7)

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. \(\frac{5}{\sqrt{27}}\)

Explain why proposed solutions of radical equations must be checked.

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(-1) $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{b} \cdot \sqrt[5]{4 a} $$

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (9.6,2.5) and (-1.9,-3.7)

Find the perimeter and area of the trapezoid. (Hint: The area of a trapezoid is the product of half the height \(6 \sqrt{3}\) meters and the sum of the bases \(2 \sqrt{63}\) and \(7 \sqrt{7}\) meters.)

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ g(-19) $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 75 $$

Find the midpoint of each line segment whose endpoints are given. (6,-8)\(;(2,4)\)

a. Add: \(\sqrt{3}+\sqrt{3}\) b. Multiply: \(\sqrt{3} \cdot \sqrt{3}\) c. Describe the differences in parts (a) and (b).

$$ \text { Solve: } \sqrt{\sqrt{x+3}+\sqrt{x}}=\sqrt{3} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(3) $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 20 $$

Find the midpoint of each line segment whose endpoints are given. (3,9)\(;(7,11)\)

a. Add: \(2 \sqrt{5}+\sqrt{5}\) b. Multiply: \(2 \sqrt{5} \cdot \sqrt{5}\) c. Describe the differences in parts (a) and (b).

The cost \(C(x)\) in dollars per day to operate a small delivery service is given by \(C(x)=80 \sqrt[3]{x}+500,\) where \(x\) is the number of deliveries per day. In July, the manager decides that it is necessary to keep delivery costs below \(\$ 1620.00 .\) Find the greatest number of deliveries this company can make per day and still keep overhead below \(\$ 1620.00\)

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(2) $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 48 $$

Find the midpoint of each line segment whose endpoints are given. (-2,-1)\(;(-8,6)\)

$$ \text { Multiply: }(\sqrt{2}+\sqrt{3}-1)^{2} $$

Write each expression in the form \(a+b i .\) $$ i^{3}+i^{4} $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ g(1) $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 45 $$

Find the midpoint of each line segment whose endpoints are given. (-3,-4)\(;(6,-8)\)

$$ \text { Multiply: }(\sqrt{5}-\sqrt{2}+1)^{2} $$

Write each expression in the form \(a+b i .\) $$ i^{8}-i^{7} $$

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}+1 $$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 3, because 8 is a perfect cube. $$ 16 $$

Find the midpoint of each line segment whose endpoints are given. (6,3)\(;(-1,-3)\)

Write each expression in the form \(a+b i .\) $$ i^{6}+i^{8} $$

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}-2 $$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 3, because 8 is a perfect cube. $$ 56 $$

Find the midpoint of each line segment whose endpoints are given. (-2,5)\(;(2,6)\)

Write each expression in the form \(a+b i .\) $$ i^{4}+i^{12} $$

Identify the domain and then graph each function. $$ g(x)=\sqrt[3]{x-1} $$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 3, because 8 is a perfect cube. $$ 54 $$

Find the midpoint of each line segment whose endpoints are given. \(\left(\frac{1}{2}, \frac{3}{8}\right) ;\left(-\frac{3}{2}, \frac{5}{8}\right)\)

Write each expression in the form \(a+b i .\) $$ 2+\sqrt{-9} $$

Identify the domain and then graph each function. $$ g(x)=\sqrt[3]{x+1} $$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 3, because 8 is a perfect cube. $$ 80 $$

Find the midpoint of each line segment whose endpoints are given. \(\left(-\frac{2}{5}, \frac{7}{15}\right) ;\left(-\frac{2}{5},-\frac{4}{15}\right)\)