1. \( \sqrt{12} \times \sqrt{15}= \) a. 5 b. \( 5 \sqrt{6} \) c. \( 6 \sqrt{5} \) d. 6 2. Which of the following is equal to ' \( x \) '? a. \( \sqrt[12]{\left(x^{4}\right)^{\frac{1}{3}}} \) b. \( \boldsymbol{x}^{\frac{12}{7}} \times \boldsymbol{x}^{\frac{7}{12}} \) c. \( \left(\sqrt{x^{3}}\right)^{\frac{2}{3}} \) d. \( x^{\frac{12}{7}}-x^{\frac{5}{7}} \) 3. An irrational number between 5 and 6 is a. none of these b. \( \sqrt{5+6} \) c. \( \sqrt{5-6} \) c. \( \sqrt{5} \times 6 \) 4. \( 2 \sqrt{3}+\sqrt{3} \) is equal to a. \( 2 \sqrt{6} \) b. \( 3 \sqrt{6} \) 3|Pag e c. 3 d. \( 3 \sqrt{3} \) 5. If \( x=3+\sqrt{8} \), then the value of \( \left(x^{2}+\frac{1}{x^{2}}\right) \) is a. 32 b. 34 c. 6 d. 12 6. Fill in the blanks: The two rational number which are their own multiplicative inverses are 7. Fill in the blanks: The value of \( \frac{2}{\sqrt{5}-\sqrt{3}} \) is . \( \qquad \) 8. Find: \( 125^{\frac{-1}{3}} \) 9. Identify whether \( \sqrt{45} \) is rational number or irrational number. 10. Write the decimal form and state its kind of decimal expansion. \( \frac{36}{100} \) 11. Simplify the expression: \( (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) \)
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\( \sqrt{12} \times \sqrt{15} \) can be simplified as \( \sqrt{12 \times 15} \). \( 12 \times 15 = 180 \), so \( \sqrt{12 \times 15} = \sqrt{180} \). \( \sqrt{180} \) can be further simplified as \( \sqrt{36 \times 5} \). \( \sqrt{36 \times 5} = Show more…
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a. If $b^{2}=a,$ then _______ is a square root of _______. b. Given $a>0,$ the symbol $\sqrt{a}$ denotes the positive or _______ square root of a. c. $b$ is an $n$ th root of $a$ if _______ = _______. d. Given the symbol $\sqrt[n]{a},$ the value $n$ is called the _______ and $a$ is called the _______. e. The symbol $\sqrt[3]{a}$ denotes the _______ root of $a$. f. The expression $\sqrt{-4}$ (is/is not) a real number. The expression $-\sqrt{4}$ (isfis not) a real number. g. The expression $\sqrt[n]{a^{n}}=|a|$ if $n$ is (even/odd). The expression $\sqrt[n]{a^{n}}=a$ if $n$ is (even/odd). h. Given a right triangle with legs $a$ and $b$ and hypotenuse $c,$ the _______ theorem is stated as $a^{2}+b^{2}=$ _______. i. In interval notation, the domain of $f(x)=\sqrt{x}$ is _______, whereas the domain of $g(x)=\sqrt[3]{x}$ is _______. j. Which of the following values of $x$ are not in the domain of $h(x)=\sqrt{x+3}: \quad x=-5, x=-4, x=-3, x=-2, x=-1, x=0 ?$
Radicals and Complex Numbers
Definition of an $n$ th Root
Which of the following are rational and which are irrational? (a) $-\sqrt{9}$ (b) 0.375 (c) $(3 \sqrt{2})(5 \sqrt{2})$ (d) $(1+\sqrt{3})^{2}$
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