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What is the mass of an object if the force remains the same at 225 N and the acceleration is doubled to 10 m/s²?
What is the mass of an object if the force remains the same at 225 N and the acceleration is doubled to 10 m/s²?
Which statement correctly describes the simplification of the expression $\left( \left( \frac{4x^{-3}y^{-4}}{7x^{-5}y^{0}} \right)^{0} \right)^{-1}$?
Which statement correctly describes the simplification of the expression $\left( \left( \frac{4x^{-3}y^{-4}}{7x^{-5}y^{0}} \right)^{0} \right)^{-1}$?
How should you calculate the distance from Yor's house to Anya's school given the description of their locations?
How should you calculate the distance from Yor's house to Anya's school given the description of their locations?
What is the distance from Yor's house to Anya's school given that Yor walked 14 km south and then 8 km west?
What is the distance from Yor's house to Anya's school given that Yor walked 14 km south and then 8 km west?
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Which student's statement about zero and negative exponents is true?
Which student's statement about zero and negative exponents is true?
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What should be the outcome of simplifying $\left( 64x^{8} \right)^{\frac{1}{6}}$?
What should be the outcome of simplifying $\left( 64x^{8} \right)^{\frac{1}{6}}$?
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What does negative exponent notation signify when applied to a base number?
What does negative exponent notation signify when applied to a base number?
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What is the correct interpretation of the term 'the power of a power' in exponent rules?
What is the correct interpretation of the term 'the power of a power' in exponent rules?
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What is the simplified form of the expression $\frac{a^{\frac{3}{2}}*a^{\frac{5}{4}}}{a^{\frac{1}{4}}}$?
What is the simplified form of the expression $\frac{a^{\frac{3}{2}}*a^{\frac{5}{4}}}{a^{\frac{1}{4}}}$?
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What is the result of simplifying $\left( \frac{16x^{12}}{81y^{4}} \right)^{\frac{3}{4}}$?
What is the result of simplifying $\left( \frac{16x^{12}}{81y^{4}} \right)^{\frac{3}{4}}$?
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What is the final answer Keith should arrive at when simplifying $\frac{3m^{-4}n^{2}}{15m^{-3}n^{-2}}$?
What is the final answer Keith should arrive at when simplifying $\frac{3m^{-4}n^{2}}{15m^{-3}n^{-2}}$?
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What is the MOST efficient method for simplifying $\frac{\sqrt{75x^{4}}}{\sqrt{3x}}$?
What is the MOST efficient method for simplifying $\frac{\sqrt{75x^{4}}}{\sqrt{3x}}$?
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Is the expression $\sqrt{\frac{8}{7}}$ simplified already?
Is the expression $\sqrt{\frac{8}{7}}$ simplified already?
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What is the simplified expression of $\sqrt[4]{128a^{12}b^{17}}$?
What is the simplified expression of $\sqrt[4]{128a^{12}b^{17}}$?
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Which approach is more efficient when simplifying $\sqrt[4]{81x^{12}y^{8}}$?
Which approach is more efficient when simplifying $\sqrt[4]{81x^{12}y^{8}}$?
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What type of variation is occurring when one variable increases while the other decreases?
What type of variation is occurring when one variable increases while the other decreases?
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How does the cost of a product M relate to the price of production N if M varies directly as N?
How does the cost of a product M relate to the price of production N if M varies directly as N?
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What type of variation is demonstrated by the relationship of profit with the number of items sold and the selling price per item?
What type of variation is demonstrated by the relationship of profit with the number of items sold and the selling price per item?
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Which student correctly translated the formula y = \frac{kx}{z} into words?
Which student correctly translated the formula y = \frac{kx}{z} into words?
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Based on the table values provided, how does N relate to M?
M 5 4 3 2 1
N 25 20 15 10 5
Based on the table values provided, how does N relate to M? M 5 4 3 2 1 N 25 20 15 10 5
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If a force of 150 N moves a 10 kg object with an acceleration of 3 m/s², what force would be needed to move a different object at the same acceleration?
If a force of 150 N moves a 10 kg object with an acceleration of 3 m/s², what force would be needed to move a different object at the same acceleration?
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What happens to the acceleration of an object if the applied force is doubled while keeping the mass constant?
What happens to the acceleration of an object if the applied force is doubled while keeping the mass constant?
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What type of variation is represented when a variable is directly affected by two other variables, one directly and the other inversely?
What type of variation is represented when a variable is directly affected by two other variables, one directly and the other inversely?
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What is the first step Nezuko should take to simplify the expression $\sqrt{3}(\sqrt{6} - 2\sqrt{6})$?
What is the first step Nezuko should take to simplify the expression $\sqrt{3}(\sqrt{6} - 2\sqrt{6})$?
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Which is the correct order of steps to simplify the expression $\frac{\sqrt{98x^{3}y^{5}}}{\sqrt{2x}} - 3\sqrt{x^{5}y}$?
Which is the correct order of steps to simplify the expression $\frac{\sqrt{98x^{3}y^{5}}}{\sqrt{2x}} - 3\sqrt{x^{5}y}$?
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What happens to the expression $- 3\sqrt{18} + 3\sqrt{8} - \sqrt{24}$ after simplification?
What happens to the expression $- 3\sqrt{18} + 3\sqrt{8} - \sqrt{24}$ after simplification?
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Is Tanjiro correct in his procedure for simplifying the expression $\frac{\sqrt{3} + 2}{\sqrt{3} - 2}$ resulting in $- 7 - 4\sqrt{3}$?
Is Tanjiro correct in his procedure for simplifying the expression $\frac{\sqrt{3} + 2}{\sqrt{3} - 2}$ resulting in $- 7 - 4\sqrt{3}$?
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Was the architect's process for calculating the area of the circular garden correct when using $\pi r^2$ with $r = \frac{5\sqrt{2}}{2}$?
Was the architect's process for calculating the area of the circular garden correct when using $\pi r^2$ with $r = \frac{5\sqrt{2}}{2}$?
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If the area of the playground is 64 square meters, what should be the first step to find the length of the mesh net required?
If the area of the playground is 64 square meters, what should be the first step to find the length of the mesh net required?
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After simplifying $\sqrt{3}(\sqrt{27} - \sqrt{3})$, what is the simplified form?
After simplifying $\sqrt{3}(\sqrt{27} - \sqrt{3})$, what is the simplified form?
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In the expression $\frac{\sqrt{18} - \sqrt{8}}{\sqrt{2}}$, what should be done first to simplify it?
In the expression $\frac{\sqrt{18} - \sqrt{8}}{\sqrt{2}}$, what should be done first to simplify it?
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Study Notes
Mathematics 9 - Variation & Equations
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Variation Types:
- Direct Variation: One variable increases as the other increases, or one decreases as the other decreases. A directly proportional relationship is represented by the equation y = kx.
- Inverse Variation: One variable increases as the other decreases, or one decreases as the other increases. An inversely proportional relationship is represented by the equation y = k/x.
- Joint Variation: More than two variables are involved. One variable changes in proportion to the product of other variables.
- Combined Variation: A combination of direct and inverse variation.
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Relationships between Variables:
- The cost of a product varies directly with the price of production.
- The profit varies jointly with the number of items sold and the selling price per item.
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Formulas & Statements:
- y = kx/z: y is directly proportional to x and inversely proportional to z.
- y is directly proportional to x and z: y = kxz.
- N varies directly as M: N = 5M.
- N varies inversely as M: NM= 5
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Simplifying Expressions
- Use order of operations (PEMDAS/BODMAS) when simplifying expressions.
- Simplify expressions using laws of exponents: xa * xb = xa+b; xa / xb = xa-b; (xa)b = xab*. Be careful with negative exponents.
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Problem Solving: Word problems often translate into equations. Identify the variables and their relationships (i.e. directly, inversely, or jointly) to find an appropriate equation. Solve for the unknown variables.
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Example Problems
- Calculating rates: Given force, mass and acceleration to calculate mass, if acceleration or force changes.
- Simplifying expressions such as: -3√18+3√8-√24
- Solving for missing sides.
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Special Cases:
- A variable raised to the zero power equals 1. This should be highlighted.
- A variable raised to a negative power is reciprocated.
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Description
Explore the different types of variation in mathematics through this quiz, focusing on direct, inverse, joint, and combined variation. Understand how these concepts relate to real-world scenarios and their mathematical representations with formulas. Test your knowledge and enhance your skills in this critical area of 9th-grade mathematics.
