Questions

Simplify \( 5\sqrt{96} - 7\sqrt{54} \)

Keywords: simplify radicals, combining like radicals, square roots, high school math
Step-by-step solution to simplifying \( 5\sqrt{96} - 7\sqrt{54} \) by factoring radicands and combining like radicals.

Lake Water Level Decreasing 3% Weekly: Find Level After 8 Weeks

Keywords: exponential decay, percent decrease, lake water level, high school math
Step-by-step solution to find lake water level after 8 weeks of 3% weekly decrease using the exponential decay formula.

Expand \( \log_2 \frac{x^5}{8} \) Using Logarithmic Properties

Keywords: expand logarithm, logarithmic properties, quotient rule, power rule, high school math
Step-by-step expansion of \( \log_2 \frac{x^5}{8} \) using the quotient and power rules of logarithms.

Solve \( -4 \ln(6x) = 2 \) Using Natural Logarithm Inverse Property

Keywords: natural logarithm, logarithmic equation, inverse property, high school math
Step-by-step solution to solve \( -4 \ln(6x) = 2 \) by isolating the logarithm and using the inverse property of \( e \) and \( \ln \).

Discontinuities of Rational Function \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \)

Keywords: rational function, discontinuities, vertical asymptote, hole, factoring polynomials, high school math
Step-by-step analysis of discontinuities in \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \) by factoring numerator and denominator.

Subtract and Simplify \( \frac{5x}{3x^2} - \frac{7}{6x} \)

Keywords: subtract rational expressions, least common denominator, simplify fractions, high school math
Step-by-step solution to subtracting \( \frac{5x}{3x^2} - \frac{7}{6x} \) by finding LCD and simplifying the result.

Solve \( \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} \) and Check for Extraneous Solution

Keywords: rational equation, extraneous solution, factoring trinomials, LCD, high school math
Step-by-step solution to solving the rational equation \( \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} \) and checking for extraneous solutions.

Express tan B as a Simplest Fraction in a Right Triangle

Keywords: tan B, right triangle, tangent ratio, simplest fraction, radical, adjacent side, opposite side
This problem involves finding \( \tan B \) in a right triangle (right-angled at \( C \)) using the tangent ratio (opposite/adjacent). The solution identifies the sides opposite and adjacent to \( \angle B \), applies the tangent formula, and simplifies the fraction.

Leading Coefficient and Degree of Polynomial -9u⁴ + 12u⁸ - 12 + 5u

Keywords: leading coefficient, degree of polynomial, polynomial terms, monomial degree, algebra
This problem calculates the leading coefficient (12) and degree (8) of the polynomial \( -9u^4 + 12u^8 - 12 + 5u \) by analyzing term degrees and the leading term’s coefficient.

High School Math - Hinge Theorem to Solve for x in Angle Inequality

Keywords: Hinge Theorem, SAS Inequality, angle inequality, solving for x, high school math
This problem uses the Hinge Theorem (SAS Inequality) to determine the inequality for \(x\), given two triangles with congruent sides (length 7) and different included angles (\(10^\circ\) and \(\frac{1}{8}x^\circ\)) and third sides (13 and 21). The solution is \(x > 80\).