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Home Question Bank Math Algebra Archive 2025 January 05

Algebra Questions from Jan 05,2025

Browse the Algebra Q&A Archive for Jan 05,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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\#9: Combine into a single logarithm \( \log _{3} A+\log _{3} B-3 \log _{3} C \). \( (4 \) Points) Expand each as much as possible. \#6: \( \log _{3}(3 x) \) \( \# 7: \log _{2}(x(x+1)) \) \( \# 8: \log \left(x^{\wedge} 5^{*} y^{\wedge} 6 / z^{\wedge} 4\right) \) Expand each as much as possible. (3 Points Each) \#6: \( \log _{3}(3 x) \) \#7: \( \log _{2}(x(x+1)) \) \#8: \( \log _{\left(x^{\wedge} 5^{\star} y^{\wedge} 6 / z^{\wedge} 4\right)} \) 26. \( 2 x^{2}+5 x-3 \) 27. \( 2 x^{2}-8 x-10 \) The first two terms of arithmetic sequences are shown in each case below. Determine the value of the requested term. Show the work that leads to your answer. \( \begin{array}{ll}\text { a) } a_{1}=7 \text { and } a_{2}=20 \text {, find } a_{2 s} & \text { (b) } b_{1}=9 \text { and } b_{2}=4, \text { find } b_{30}\end{array} \) N-GEN MATH ALGEBRA I - UNIT 4 - LINEAR FUNCTIONS-LESSON 14 EMATHINstruction, RED HOOK, NY 12571, O2022 \#2: Graph \( y=2+2^{x-2} \) using \( x=0,1,2,3,4 \) (Count by 1 's up to 8 along the \( y \)-axis) (5 Points) 1. Simplify the following without the use of a calculator: (a) \( \left(\frac{1}{2}\right)^{x-1} \cdot(\sqrt{2})^{2 x-1} \) (b) \( \frac{2.3^{x}-3^{x-2}}{2.3^{x-2}} \) (c) \( \frac{6^{x}-3^{x-2}}{6.18^{x-2}} \) (d) \( (\sqrt{2}-3)(\sqrt{8}+2) \) (e) \( \frac{\sqrt{20 x^{5}}-\sqrt{10 x}+\sqrt{2 x^{6}}}{\sqrt{2} x^{6}} \) (f) \( \frac{\sqrt[4]{16 x^{5}}}{\sqrt{\sqrt{x}}} \) (g) \( \sqrt{72}-\sqrt{2^{7}} \) 2. Simplify the following: \( \frac{2}{\sqrt{3}}-\frac{\sqrt{3}}{2} \) (rationalise the denominator) 3. Solve for \( x \) if: \( 3^{3 x}+3^{3 x}+3^{3 x}=3^{9 x} \) 4. Solve the following equations: (a) \( x^{\frac{1}{3}}=9 \) (b) \( \quad\left(\frac{1}{3}\right)^{x}=9 \) (c) \( \frac{1}{3} x=9 \) (d) \( \frac{1}{16} x^{\frac{3}{4}}=4 \) (e) \( \frac{1}{16} \cdot 2^{\frac{3}{4} x}=4 \) (f) \( 4 x^{\frac{2}{3}}+2=18 \) 5. Solve for \( x \) : (a) \( 2^{x+1}+2^{x+2}=6 \) (b) \( \quad 2^{x+1} \cdot 2^{x+2}=4 \) 6. Solve for \( x \) : (a) \( \quad 2^{2 x}+3.2^{x}-4=0 \) (b) \( 4^{x}-2^{x-2}=0 \) (c) \( x^{\frac{2}{3}}+4 x^{\frac{1}{3}}-5=0 \) (d) \( 3^{x}+3.3^{-x}-4=0 \) SECTION B (9 MARKS) 1. Show that \( (\sqrt{3})^{-2 x}+4.3^{1-x}=\frac{13}{3^{x}} \quad \) 2. Show that \( \frac{9^{x+1}-6.3^{2 x}}{(\sqrt{3})^{4 x+1}}=\sqrt{3} \) 3. Show that \( \left(\frac{\sqrt{x^{5}}-\sqrt{x^{3}}}{x}\right)^{2}=x(x-1)^{2}, x>0 \) 4. Calculate 4. \( (\sqrt{2})^{3} \cdot 8^{\frac{2}{3}} \) without the use of a calculator (simplest surd form). 5. Given: \( \mathrm{P}=\frac{5^{x}+5^{x}+5^{x}+5^{x}+5^{x}}{5^{x+5}} \). If \( x=2011 \) then, (1) \( \mathrm{P}=5^{8039} \) (2) \( \mathrm{P}=1 \) (3) \( \mathrm{P}=5^{-4} \) (4) \( \mathrm{P}=5 \) 1. Simplify the following without the use of a calculator: (a) \( \left(\frac{1}{2}\right)^{x-1} \cdot(\sqrt{2})^{2 x-1} \) (b) \( \frac{2.3^{x}-3^{x-2}}{2.3^{x-2}} \) (c) \( \frac{6^{x}-3^{x-2}}{6.18^{x-2}} \) (d) \( (\sqrt{2}-3)(\sqrt{8}+2) \) (e) \( \frac{\sqrt{20 x^{5}}-\sqrt{10 x}+\sqrt{2 x^{6}}}{\sqrt{2} x^{6}} \) (f) \( \frac{\sqrt[4]{16 x^{5}}}{\sqrt{\sqrt{x}}} \) (g) \( \sqrt{72}-\sqrt{2^{7}} \) 2. Simplify the following: \( \frac{2}{\sqrt{3}}-\frac{\sqrt{3}}{2} \) (rationalise the denominator) 3. Solve for \( x \) if: \( 3^{3 x}+3^{3 x}+3^{3 x}=3^{9 x} \) 4. Solve the following equations: (a) \( x^{\frac{1}{3}}=9 \) (b) \( \left(\frac{1}{3}\right)^{x}=9 \) (c) \( \quad \frac{1}{3} x=9 \) (d) \( \frac{1}{16} x^{\frac{3}{4}}=4 \) (e) \( \frac{1}{16} \cdot 2^{\frac{3}{4} x}=4 \) (f) \( 4 x^{\frac{2}{3}}+2=18 \) 5. Solve for \( x \) : (a) \( 2^{x+1}+2^{x+2}=6 \) (b) \( \quad 2^{x+1} \cdot 2^{x+2}=4 \) 6. Solve for \( x \) : (a) \( 2^{2 x}+3.2^{x}-4=0 \) (b) \( 4^{x}-2^{x-2}=0 \) (c) \( x^{\frac{2}{3}}+4 x^{\frac{1}{3}}-5=0 \) (d) \( 3^{x}+3.3^{-x}-4=0 \) SECTION B (9 MARKS) 1. Show that \( (\sqrt{3})^{-2 x}+4.3^{1-x}=\frac{13}{3^{x}} \quad \) 2. Show that \( \frac{9^{x+1}-6.3^{2 x}}{(\sqrt{3})^{4 x+1}}=\sqrt{3} \) 3. Show that \( \left(\frac{\sqrt{x^{5}}-\sqrt{x^{3}}}{x}\right)^{2}=x(x-1)^{2}, x>0 \) 4. Calculate 4. \( (\sqrt{2})^{3} \cdot 8^{\frac{2}{3}} \) without the use of a calculator (simplest surd form). 5. Given: \( \mathrm{P}=\frac{5^{x}+5^{x}+5^{x}+5^{x}+5^{x}}{5^{x+5}} \). If \( x=2011 \) then, (1) \( \mathrm{P}=5^{8039} \) (2) \( \mathrm{P}=1 \) (3) \( \mathrm{P}=5^{-4} \) (4) \( \mathrm{P}=5 \) \#9: Find a polynomial of degree 4 with zeroes of 0 and 3 , where 3 has a multiplicity of 3 , and \( P(1)=24 \). (7 Points)
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