Chapter 5

Find, to the nearest square centimeter, the area of a triangle that measures 14 centimeters by 16 centimeters by 18 centimeters.

For Problems \(75-84\), express each of the following as a single fraction involving positive exponents only. \(x^{-3}-y^{-1}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{72 x^{\frac{3}{4}}}{6 x^{\frac{1}{2}}}\right)^{2}\)

Find, to the nearest square yard, the area of a triangular plot of ground that measures 45 yards by 60 yards by 75 yards.

For Problems \(75-84\), express each of the following as a single fraction involving positive exponents only. \(2 x^{-1}-3 y^{-2}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{60 a^{\frac{1}{5}}}{15 a^{\frac{3}{4}}}\right)^{2}\)

How would you simplify the expression \(\frac{\sqrt{8}+\sqrt{12}}{\sqrt{2}}\) ?

Do the following problems, where the variable could be any real number as long as the radical represents a real number. Use absolute-value signs in the answers as necessary. (a) \(\sqrt{125 x^{2}}\) (b) \(\sqrt{16 x^{4}}\) (c) \(\sqrt{8 b^{3}}\) (d) \(\sqrt{3 y^{5}}\) (e) \(\sqrt{288 x^{6}}\) (f) \(\sqrt{28 m^{8}}\) (g) \(\sqrt{128 c^{10}}\) (h) \(\sqrt{18 d^{7}}\) (i) \(\sqrt{49 x^{2}}\) (j) \(\sqrt{80 n^{20}}\) (k) \(\sqrt{81 h^{3}}\)

Find the area of an equilateral triangle, each of whose sides is 18 inches long. Express the area to the nearest square inch.

Express each of the following as a single fraction involving positive exponents only. \(3 a^{-2}+4 b^{-1}\)

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{64 a^{\frac{1}{3}}}{16 a^{\frac{5}{9}}}\right)^{3}\)

Express each of the following as a single fraction involving positive exponents only. \(a^{-1}+a^{-1} b^{-3}\)

Why is \(\sqrt{-9}\) not a real number?

Express each of the following as a single fraction involving positive exponents only. \(x^{-1} y^{-2}-x y^{-1}\)

For Problems \(81-90\), perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\sqrt{2} \sqrt[4]{2}\)

Why is it that we say 25 has two square roots \((5\) and \(-5)\), but we write \(\sqrt{25}=5\) ?

Express each of the following as a single fraction involving positive exponents only. \(x^{2} y^{-2}-x^{-1} y^{-3}\)

How is the multiplication property of 1 used when simplifying radicals?

Express each of the following as a single fraction involving positive exponents only. \(2 x^{-1}-3 x^{-2}\)

How could you find a whole number approximation for \(\sqrt{2750}\) if you did not have a calculator or table available?

Express each of the following as a single fraction involving positive exponents only. \(5 x^{-2} y+6 x^{-1} y^{-2}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{3}}{\sqrt[4]{3}}\)

Use your calculator to find a rational approximation, to the nearest thousandth, for (a) through (i). (a) \(\sqrt{2}\) (b) \(\sqrt{75}\) (c) \(\sqrt{156}\) (d) \(\sqrt{691}\) (e) \(\sqrt{3249}\) (f) \(\sqrt{45,123}\) (g) \(\sqrt{0.14}\) (h) \(\sqrt{0.023}\) (i) \(\sqrt{0.8649}\)

Is the following simplification process correct? $$ \left(3^{-2}\right)^{-1}=\left(\frac{1}{3^{2}}\right)^{-1}=\left(\frac{1}{9}\right)^{-1}=\frac{1}{\left(\frac{1}{9}\right)^{1}}=9 $$ Could you suggest a better way to do the problem?

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt{2}}{\sqrt[3]{2}}\)

Sometimes a fairly good estimate can be made of a radical expression by using whole number approximations. For example, \(5 \sqrt{35}+7 \sqrt{50}\) is approximately \(5(6)+7(7)=79\). Using a calculator, we find that \(5 \sqrt{35}+7 \sqrt{50}=79.1\), to the nearest tenth. In this case our whole number estimate is very good. For (a) through (f), first make a whole number estimate, and then use your calculator to see how well you estimated. (a) \(3 \sqrt{10}-4 \sqrt{24}+6 \sqrt{65}\) (b) \(9 \sqrt{27}+5 \sqrt{37}-3 \sqrt{80}\) (c) \(12 \sqrt{5}+13 \sqrt{18}+9 \sqrt{47}\) (d) \(3 \sqrt{98}-4 \sqrt{83}-7 \sqrt{120}\) (e) \(4 \sqrt{170}+2 \sqrt{198}+5 \sqrt{227}\) (f) \(-3 \sqrt{256}-6 \sqrt{287}+11 \sqrt{321}\)

Explain how to simplify \(\left(2^{-1} \cdot 3^{-2}\right)^{-1}\) and also how simplify \(\left(2^{-1}+3^{-2}\right)^{-1}\).

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{8}}{\sqrt[4]{4}}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt{9}}{\sqrt[3]{3}}\)

Use a calculator to simplify each of the following numerical expressions. Express your answers to the nearest hundredth. (a) \(\left(2^{-3}+3^{-3}\right)^{-2}\) (b) \(\left(4^{-3}-2^{-1}\right)^{-2}\) (c) \(\left(5^{-3}-3^{-5}\right)^{-1}\) (d) \(\left(6^{-2}+7^{-4}\right)^{-2}\) (e) \(\left(7^{-3}-2^{-4}\right)^{-2}\) (f) \(\left(3^{-4}+2^{-3}\right)^{-3}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[4]{27}}{\sqrt{3}}\)

Your friend keeps getting an error message when evaluating \(-4^{\frac{5}{2}}\) on his calculator. What error is he probably making?

Explain how you would evaluate \(27^{\frac{2}{3}}\) without a calculator.

Use your calculator to evaluate each of the following. (a) \(\sqrt[3]{1728}\) (b) \(\sqrt[3]{5832}\) (c) \(\sqrt[4]{2401}\) (d) \(\sqrt[4]{65,536}\) (e) \(\sqrt[5]{161,051}\) (f) \(\sqrt[5]{6,436,343}\)

Definition \(5.7\) states that $$ b^{\frac{m}{n}}=\sqrt[n]{b^{m}}=(\sqrt[n]{b})^{m} $$ Use your calculator to verify each of the following. (a) \(\sqrt[3]{27^{2}}=(\sqrt[3]{27})^{2}\) (b) \(\sqrt[3]{8^{5}}=(\sqrt[3]{8})^{5}\) (c) \(\sqrt[4]{16^{3}}=(\sqrt[4]{16})^{3}\) (d) \(\sqrt[3]{16^{2}}=(\sqrt[3]{16})^{2}\) (e) \(\sqrt[5]{9^{4}}=(\sqrt[5]{9})^{4}\) (f) \(\sqrt[3]{12^{4}}=(\sqrt[3]{12})^{4}\)

Use your calculator to evaluate each of the following. (a) \(16^{\frac{5}{2}}\) (b) \(25^{\frac{7}{2}}\) (c) \(16^{\frac{9}{4}}\) (d) \(27^{\frac{5}{3}}\) (e) \(343^{\frac{2}{3}}\) (f) \(512^{\frac{4}{3}}\)

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{5}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{5}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)