Questions and Answers
What is the result of using the multiplicative property of equality with signed fractions on the equation 9 = -3/4y?
What is the simplified form of the ratio 30y^4/5y^7?
6/y^3
What is the value of x in the equation -6(3x-1)+4x = 1+2(2x-5)?
5/6
A^0 = 1 is true.
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What is the simplified form of the fraction 56(u-7)(2u-3) / 7(2u-3)(u+8)?
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What is the result of sqrt(a) * sqrt(b)?
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What is the result of 3/20 divided by (-21/16)?
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What values of w satisfy the equation (w + 7)(w - 8) = 0?
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What is the product of the values p and q in the factoring of 14x^2 + 9x - 5?
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What property is used in evaluating functions?
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What is the final value when simplifying 2 * sqrt(72) * sqrt(98)?
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Simplify the expression 3x - 4(4y - 3x) + 4y.
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What is the simplified form of sqrt(2) / sqrt(3)?
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What is the value of w in the equation 3/2w - 1/7 = -8/5?
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What is the simplified ratio of 8w^2 - 12w / 4w^2 - 16?
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What is the degree and leading coefficient of the polynomial -5 - 10w + w^6?
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What does the LCM of 12a^4 and 9a^2 equal?
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Study Notes
Multiplicative Property of Equality
- If ( 9 = -\frac{3}{4}y ), then multiplying both sides by the reciprocal (-\frac{4}{3}) leads to solving ( -12 = y ).
Simplifying Ratios of Monomials
- For ( \frac{30y^4}{5y^7} ), divide coefficients and subtract exponents: ( \frac{30}{5} = 6 ) and ( y^{4-7} = \frac{1}{y^3} ), resulting in ( \frac{6}{y^3} ).
Solving Linear Equations with Variables on Both Sides
- The equation ( -6(3x-1) + 4x = 1 + 2(2x-5) ) requires the distributive property, combining like terms, and isolating ( x ) to find ( x = \frac{5}{6} ).
Evaluating Expressions with Exponents
- Any number raised to the power of zero equals one, ( a^0 = 1 ).
Simplifying Ratios of Factored Polynomials
- For ( \frac{56(u-7)(2u-3)}{7(2u-3)(u+8)} ), cancel the common factor ( (2u-3) ) to simplify to ( \frac{8(u-7)}{u+8} ).
Basic Square Root Multiplication
- The product property states ( \sqrt{a} \sqrt{b} = \sqrt{ab} ), simplifying gives results like ( 4\sqrt{3} ) from ( \sqrt{48} ).
Signed Fraction Division
- For ( \frac{3}{20} \div \left(-\frac{21}{16}\right) ), take the reciprocal, cancel common factors leading to ( -\frac{4}{35} ).
Solving Equations in Factored Form
- In ( (w + 7)(w - 8) = 0 ), set each factor to zero: solutions are ( w = -7 ) and ( w = 8 ).
Factoring Quadratics with Leading Coefficient Greater Than 1
- For ( 14x^2 + 9x - 5 ), find integers ( p ) and ( q ) such that ( pq = -70 ) (from ( 14 \times -5 )) and ( p + q = 9 ).
Evaluating Functions
- Use substitution property in functions involving linear, quadratic, or cubic equations to analyze outputs.
Advanced Square Root Multiplication
- To simplify ( 2\sqrt{72} \sqrt{98} ), first simplify each square root, leading to answer ( 168 ) through ( \sqrt{7056} ).
Simplifying Multivariate Expressions
- With ( 3x - 4(4y - 3x) + 4y ), apply distributive property, combine like terms, and simplify to yield ( 15x - 12y ).
Rationalizing Denominators
- For ( \frac{\sqrt{2}}{\sqrt{3}} ), multiply numerator and denominator by ( \sqrt{3} ) to simplify the result to ( \frac{\sqrt{6}}{3} ).
Solving Two-Step Equations with Signed Fractions
- With ( \frac{3}{2}w - \frac{1}{7} = -\frac{8}{5} ), eliminate fractions using LCD, apply the distributive property, and isolate ( w ) leading to ( w = -\frac{34}{35} ).
Simplifying Ratios of Polynomials Using GCF
- Reducing ( \frac{8w^2 - 12w}{4w^2 - 16} ) involves factoring both numerator and denominator to arrive at ( \frac{2w - 3}{w - 4} ).
Degree and Leading Coefficient of a Polynomial
- The polynomial (-5 - 10w + w^6) has a degree of ( 6 ) (highest exponent) and a leading coefficient of ( 1 ).
Simplifying Products with Square Roots
- In the expression ( \sqrt{7}(\sqrt{6} - 11) ), apply distributive property and the product property of square roots for simplification.
Finding the LCM of Monomials
- For monomials ( 12a^4 ) and ( 9a^2 ), the LCM is determined as ( 36a^4 ) combining the LCM of coefficients and highest powers of variables.
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Explore essential algebra concepts with these flashcards based on the Aleks system. Each card covers important topics such as the multiplicative property of equality and simplifying ratios of monomials. Perfect for reinforcing your understanding and preparing for assessments.
