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Home Question Bank Math Algebra Archive 2025 January 15

Algebra Questions from Jan 15,2025

Browse the Algebra 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. [-/2 Points] DETAILS MY NOTES LARPCALC10 2.5.069, Write the polynomial as the product of linear factors. \( g(x)=x^{3}-6 x^{2}+x+34 \) \( g(x)=\square \) List all the zeros of the function. (Enter your answers as a comma-separated list,) \( x=\square \) Need Help? Rusill SUBMIT ANSWER \( 1 \leftarrow \quad \) Simplify and express answers using positive exponents only. \( \frac{6 x^{-4} y^{-3}}{4 x^{5} y^{-5}} \) \( \frac{6 x^{-4} y^{-3}}{4 x^{5} y^{-5}}=\square \) (Use positive exponents only.) 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 \[ \begin{array}{l} \left(2^{4} x^{4}\right) \\ =\frac{3^{-3} y^{-6}}{2^{-3} x^{-3}} \\ =\frac{2^{3} x^{3}}{3^{3} y^{6}} \\ =\frac{8 x^{3}}{27 y^{6}} \end{array} \] 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) \( \quad(2,25)^{-\frac{1}{2}} \) \( (12)^{*} \quad(0, i)^{\frac{3}{2}} \) +. \( (-2 a+10 a b) \) Simplify and express answers using positive exponents only. Variables are restricted to avoid division by 0 . \( \frac{x^{-11}}{x^{-4}} \) \( \frac{x^{-11}}{x^{-4}}=\square \) (Use positive exponents only.) 8. \( \left(100 x^{6} y^{10}\right)^{\frac{1}{2}} \) oftermine whether each equation represents an inverse proportion. It so, find the constant of proportionality. \( \frac{1}{20}=x=\frac{5}{y} \) \( y+\frac{1}{7} x=\frac{1}{2} \) \( 0,1 x=\frac{5}{y} \) Simplify and express answers using positive exponents only. Variables are restricted to avoid division by 0 . \( 7 z^{-5} \) \( 7 z^{-5}=\square \) (Simplify your answer. Use positive exponents only.) 11 of 20 11. \( 14 y^{2}-x^{2}-3 x y \)
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