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Home Question Bank Math Archive 2025 January 15

Math Questions from Jan 15,2025

Browse the Math Q&A Archive for Jan 15,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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(5. Which of the following statements is true? A. \( 2 \sqrt{8} \) is between 5 and 6 . B. \( \sqrt{65} \) is between 6 and 7 . \( 5 \sqrt{3} \) is between 9 and 10 . D. \( 3 \sqrt{75} \) is between 12 and 13 . 1)) A quadrilateral has two angles that measure \( 198^{\circ} \) and \( 2^{\circ} \). The other two angles are in a ratio of \( 15: 17 \). What are the measures of those two angles? 4. [-/1 Points] DETAILS MY NOTES LARPCALC10 2.5.031. Find all real solutions of the polynomial equation. (Enter your answers as a comm \[ \begin{array}{l}x^{4}+6 x^{3}+3 x^{2}-16 x+6=0\end{array} \] Need Help? Read it Watch it SUBMIT ANSWER If a car has an acceleration function of \( a(t) = 6t \), what is its velocity function assuming it starts from rest? 0.4 Find missing angles in quadrilaterals 6 V 4 \[ \begin{array}{l} \left(2^{4} x^{4}\right) \\ =\frac{3^{-3} y^{-6}}{2^{-3} x^{-3}} \\ =\frac{2^{3} x^{3}}{3^{3} y^{6}} \\ =\frac{8 x^{3}}{27 y^{6}} \end{array} \] 26 To simplify expressions containing roots, we rewrite the roots as powers with rational exponents: EXAMPLE 3 Simplify: (a) \( \frac{\sqrt{x^{3}} \times \sqrt[3]{x^{2}}}{\sqrt[6]{x}} \) (b) \( \sqrt[3]{8 x^{2} y^{6}} \times x^{\frac{1}{3}} \) Solution (a) \( \frac{x^{\frac{3}{2}} \times x^{\frac{2}{3}}}{x^{\frac{1}{6}}} \) \[ \text { (b) } \quad\left(8 x^{2} y^{6}\right)^{\frac{1}{3}} \times x^{\frac{1}{3}} \] \[ =x^{\frac{3}{2}+\frac{2}{3}-\frac{1}{6}} \] \[ =x^{\frac{9+4-1}{6}} \] \[ =x^{\frac{12}{6}}=x^{2} \] Note: When applying roots to expressions, it is very important to distinguish between the behaviour of coefficients and exponents: \( \sqrt{16 x^{16}}=4 x^{8} \quad \) Take the square root of the coefficient, but divide the exponent by 2 . \( \sqrt[3]{27 x^{27}}=3 x^{9} \quad \) Take the cube root of the coefficient, but divide the exponent by 3 . EXERCISE 1 (a) Calculate the following, without the use of a calculator: (1) \( 25^{\frac{1}{2}} \) (2) \( 8^{\frac{2}{3}} \) (3) \( 9^{-\frac{3}{2}} \) (4) \( 32^{-\frac{2}{5}} \) (5) \( \left(\frac{1}{8}\right)^{\frac{1}{3}} \) (6) \( \left(\frac{4}{9}\right)^{\frac{3}{2}} \) (7) \( \left(\frac{27}{8}\right)^{-\frac{2}{3}} \) (8) \( \left(\frac{16}{81}\right)^{-\frac{3}{4}} \) (9) \( (0,25)^{\frac{1}{2}} \) (10) \( \quad(0,125)^{\frac{2}{3}} \) (11) \( \quad(2,25)^{-\frac{1}{2}} \) \( (12)^{*} \quad(0, i)^{\frac{3}{2}} \) Write the number \( 6 \times 10^{7} \) in decimal notation. \( 6 \times 10^{7}=\square \) (3) Terri makes a quilt using three sizes of fabric squares. The side length of each fabric square is the square root of the area. The areas of the fabric squares are 29,40 , and 62 square inches. Which could be the approximate side length of a square of the quilt? A. 2.9 in . B. 5.4 in . C. 6.7 in . D. 8.1 in . Express the number 0.053 in scientific notation. \( 0.053=\square \) (Use the multiplication symbol in the math palette as needed.) +. \( (-2 a+10 a b) \)
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