EBUS PC11 Chapter 5 Radicals Test-1

Step 1: Convert the numbers in option D to decimals.\nStep 2: \(0.\overline{33}=0.3333\ldots\).\nStep 3: \(\frac{17}{50}=0.34\).\nStep 4: Compare the decimals: \(0.3333\ldots < 0.34 < 0.3414\).\nStep 5: Therefore option D is written from smallest to largest.\nAnswer: \(0.\overline{33}, \frac{17}{50}, 0.3414\)

Step 1: Recall the definition of a rational number.\nStep 2: A rational number is any number that can be written as \(\frac{x}{y}\), where \(x\) and \(y\) are integers and \(y\neq 0\).\nStep 3: This matches the description in the question.\nAnswer: a rational number

Step 1: Convert the mixed number \(1\frac{3}{4}\) to an improper fraction.\nStep 2: \(1\frac{3}{4}=\frac{7}{4}\).\nStep 3: Since \(\frac{7}{4}\) is a ratio of two integers, it is rational.\nStep 4: Therefore the statement saying it is not rational is false.\nAnswer: \(1\frac{3}{4}\) is a real number but not a rational number.

Step 1: Use the negative exponent rule: \(a^{-m}=\frac{1}{a^m}\).\nStep 2: \((-c^3)^{-\frac{1}{4}}=\frac{1}{(-c^3)^{\frac{1}{4}}}\).\nStep 3: Use the fractional exponent rule: \(a^{\frac{1}{4}}=\sqrt[4]{a}\).\nStep 4: Therefore \((-c^3)^{\frac{1}{4}}=\sqrt[4]{-c^3}\).\nAnswer: \(\frac{1}{\sqrt[4]{-c^3}}\)

Step 1: Rewrite the cube root using a fractional exponent: \(\sqrt[3]{x^5}=x^{\frac{5}{3}}\).\nStep 2: Raise this expression to the power \(\frac{1}{2}\): \(\left(x^{\frac{5}{3}}\right)^{\frac{1}{2}}\).\nStep 3: Multiply the exponents: \(\frac{5}{3}\cdot\frac{1}{2}=\frac{5}{6}\).\nStep 4: The expression is \(x^{\frac{5}{6}}\), so \(a=5\).\nAnswer: \(5\)

Step 1: Rewrite the cube root using a fractional exponent: \(\sqrt[3]{x^7}=x^{\frac{7}{3}}\).\nStep 2: The numerator becomes \(\left(25x^{\frac{7}{3}}\right)^2\).\nStep 3: Square each factor: \(25^2x^{\frac{14}{3}}=625x^{\frac{14}{3}}\).\nStep 4: Divide by \(5x^{\frac{1}{3}}\): \(\frac{625x^{\frac{14}{3}}}{5x^{\frac{1}{3}}}\).\nStep 5: Divide coefficients: \(\frac{625}{5}=125\).\nStep 6: Subtract exponents: \(\frac{14}{3}-\frac{1}{3}=\frac{13}{3}\).\nAnswer: \(125x^{\frac{13}{3}}\)

Step 1: To move the outside coefficient into a square root, square the coefficient.\nStep 2: \(2\sqrt{8}=\sqrt{2^2\cdot 8}\).\nStep 3: Calculate inside the radical: \(2^2\cdot 8=4\cdot 8=32\).\nAnswer: \(\sqrt{32}\)

Step 1: To move the outside coefficient into a cube root, cube the coefficient.\nStep 2: \(5\sqrt[3]{4}=\sqrt[3]{5^3\cdot 4}\).\nStep 3: Calculate \(5^3=125\).\nStep 4: Multiply: \(125\cdot 4=500\).\nAnswer: \(\sqrt[3]{500}\)

Step 1: Simplify each radical: \(\sqrt{48}=\sqrt{16\cdot 3}=4\sqrt{3}\).\nStep 2: \(2\sqrt{48}=2(4\sqrt{3})=8\sqrt{3}\).\nStep 3: Simplify \(\sqrt{12}=\sqrt{4\cdot 3}=2\sqrt{3}\), so \(-6\sqrt{12}=-12\sqrt{3}\).\nStep 4: Simplify \(\sqrt{75}=\sqrt{25\cdot 3}=5\sqrt{3}\), so \(3\sqrt{75}=15\sqrt{3}\).\nStep 5: Combine like radicals: \(8\sqrt{3}-12\sqrt{3}+15\sqrt{3}-\sqrt{3}=10\sqrt{3}\).\nAnswer: \(10\)

Step 1: Simplify the first power: \((2a^6)^3=2^3a^{18}=8a^{18}\).\nStep 2: Simplify the second power: \((3a^4)^4=3^4a^{16}=81a^{16}\).\nStep 3: Multiply the coefficients: \(8\cdot 81=648\).\nStep 4: Add exponents when multiplying powers with the same base: \(a^{18}a^{16}=a^{34}\).\nAnswer: \(648a^{34}\)

Step 1: Rewrite each radical using fractional exponents: \(\sqrt{x^9}=x^{\frac{9}{2}}\) and \(\sqrt[7]{x^5}=x^{\frac{5}{7}}\).\nStep 2: Divide powers with the same base by subtracting exponents: \(x^{\frac{9}{2}}\div x^{\frac{5}{7}}=x^{\frac{9}{2}-\frac{5}{7}}\).\nStep 3: Use a common denominator of \(14\): \(\frac{9}{2}=\frac{63}{14}\) and \(\frac{5}{7}=\frac{10}{14}\).\nStep 4: Subtract: \(\frac{63}{14}-\frac{10}{14}=\frac{53}{14}\).\nStep 5: Convert back to radical form: \(x^{\frac{53}{14}}=\sqrt[14]{x^{53}}\).\nAnswer: \(\sqrt[14]{x^{53}}\)

Step 1: Treat the figure as a large rectangle minus the missing top-right rectangle.\nStep 2: The large rectangle has width \(7\sqrt{3}+5\sqrt{5}\) and height \(4\sqrt{3}+3\sqrt{5}\).\nStep 3: Its area is \((7\sqrt{3}+5\sqrt{5})(4\sqrt{3}+3\sqrt{5})\).\nStep 4: Expand: \(84+21\sqrt{15}+20\sqrt{15}+75=159+41\sqrt{15}\).\nStep 5: The missing rectangle has width \(3\sqrt{3}\). Its height is \((4\sqrt{3}+3\sqrt{5})-2\sqrt{5}=4\sqrt{3}+\sqrt{5}\).\nStep 6: Its area is \(3\sqrt{3}(4\sqrt{3}+\sqrt{5})=36+3\sqrt{15}\).\nStep 7: Subtract the missing area: \((159+41\sqrt{15})-(36+3\sqrt{15})=123+38\sqrt{15}\).\nAnswer: \(38\sqrt{15}+123\)

Step 1: Simplify the radicals: \(\sqrt{24}=2\sqrt{6}\) and \(\sqrt{18}=3\sqrt{2}\).\nStep 2: Substitute: \(\frac{5\sqrt{24}}{2\sqrt{18}}=\frac{5(2\sqrt{6})}{2(3\sqrt{2})}\).\nStep 3: Simplify: \(\frac{10\sqrt{6}}{6\sqrt{2}}=\frac{5\sqrt{6}}{3\sqrt{2}}\).\nStep 4: Divide the radicals: \(\frac{\sqrt{6}}{\sqrt{2}}=\sqrt{3}\).\nStep 5: The expression becomes \(\frac{5\sqrt{3}}{3}\), so \(b=3\).\nAnswer: \(3\)

Step 1: Simplify \(\sqrt{75}=5\sqrt{3}\), so \(8\sqrt{75}=40\sqrt{3}\).\nStep 2: The expression becomes \(\frac{40\sqrt{3}+9\sqrt{2}}{3\sqrt{10}}\).\nStep 3: Rationalize the denominator by multiplying by \(\frac{\sqrt{10}}{\sqrt{10}}\).\nStep 4: The denominator becomes \(3\sqrt{10}\cdot\sqrt{10}=30\).\nStep 5: The numerator becomes \((40\sqrt{3}+9\sqrt{2})\sqrt{10}=40\sqrt{30}+9\sqrt{20}\).\nStep 6: Simplify \(\sqrt{20}=2\sqrt{5}\), so the numerator is \(40\sqrt{30}+18\sqrt{5}\).\nStep 7: Divide numerator and denominator by \(2\): \(\frac{40\sqrt{30}+18\sqrt{5}}{30}=\frac{20\sqrt{30}+9\sqrt{5}}{15}\).\nAnswer: \(\frac{20\sqrt{30}+9\sqrt{5}}{15}\)

Step 1: Move the radical expression to the other side: \(x+1=\sqrt{x^2+3}\).\nStep 2: Since the right side is a square root, we need \(x+1\geq 0\).\nStep 3: Square both sides: \((x+1)^2=x^2+3\).\nStep 4: Expand: \(x^2+2x+1=x^2+3\).\nStep 5: Subtract \(x^2\) from both sides: \(2x+1=3\).\nStep 6: Solve: \(2x=2\), so \(x=1\).\nStep 7: Check: \(1-\sqrt{1^2+3}=1-2=-1\), so the solution is valid.\nAnswer: \(1\)

Step 1: Since \(\sqrt{x+4}\geq 0\), the right side must satisfy \(3x-12\geq 0\), so \(x\geq 4\).\nStep 2: Square both sides: \(x+4=(3x-12)^2\).\nStep 3: Expand the right side: \((3x-12)^2=9x^2-72x+144\).\nStep 4: Write the equation: \(x+4=9x^2-72x+144\).\nStep 5: Rearrange: \(9x^2-73x+140=0\).\nStep 6: Factor or use the quadratic formula to get \(x=5\) or \(x=\frac{28}{9}\).\nStep 7: Check the restrictions. \(x=\frac{28}{9}\) makes \(3x-12<0\), so it is extraneous.\nStep 8: Check \(x=5\): \(\sqrt{5+4}=3\), and \(3(5)-12=3\).\nAnswer: \(5\)

Step 1: Isolate the radical: \(\sqrt{2x+40}=x-4\).\nStep 2: Since the square root is nonnegative, a valid solution must have \(x-4\geq 0\), so \(x\geq 4\).\nStep 3: Square both sides: \(2x+40=(x-4)^2\).\nStep 4: Expand: \(2x+40=x^2-8x+16\).\nStep 5: Rearrange: \(x^2-10x-24=0\).\nStep 6: Factor: \((x-12)(x+2)=0\).\nStep 7: The possible roots are \(x=12\) and \(x=-2\).\nStep 8: Check the restriction \(x\geq 4\). The root \(x=-2\) does not satisfy this restriction, so it is extraneous.\nAnswer: \(-2\)