🧮 algebra

### Exercice 1 1) Problème : Trouver la bonne équation dont $\sqrt{5}$ est solution.

1. **Stating the problem:** We know the weight of 12 sheets of thick paper is 40 kg. We want to find how many sheets would weigh 1 g. 2. **Convert all weights to the same units:**

1. Stating the problem: Factorise the expression $$-32h^5j^3y^4 + 64h^2j^4y^7$$.\n\n2. Identify the greatest common factor (GCF) for each part:\n- Numerical coefficients: GCF of 32

1. The problem is to solve the inequality: $$3x + 4 < 8$$. 2. Subtract 4 from both sides to isolate the term with $$x$$:

1. The problem is to factorise the quadratic expression $$6x^{2} + 35x + 49$$. 2. First, identify the coefficients: $$a = 6$$, $$b = 35$$, and $$c = 49$$.

1. The user has provided a long list of numbers ending with an underscore, suggesting the need to find the correct continuation or next number in the sequence. 2. Since the given s

1. Simplify $3^4 \times 3^2$. Using the product of powers rule: $a^m \times a^n = a^{m+n}$,

1. The problem requires us to simplify the expression $a^{-9} \times a^{-3}$. 2. Recall the exponent multiplication rule: when multiplying like bases, add the exponents.

1. **State the problem:** Simplify the expression $b^{-5} \times \sqrt[5]{b^3}$. 2. **Rewrite the radical as an exponent:** Recall that $\sqrt[5]{b^3} = b^{\frac{3}{5}}$.

1. **Minimize area of a closed cylinder with volume 250\pi ml** Given volume $V = 250\pi$, volume formula: $$V = \pi r^2 h = 250\pi \implies r^2 h = 250$$

1. The problem involves analyzing the parabola given by the equation $y^2 = 8x$. 2. We identify the parabola as one that opens to the right because it is in the form $y^2 = 4ax$.

1. The problem asks to simplify the expression $b^{-5} \times \sqrt[5]{b^3}$. 2. Recall that $\sqrt[5]{b^3}$ can be written as $b^{\frac{3}{5}}$.

1. Problem: Simplify expression 8.1: $\frac{(\log 3 - \log 5)(\log 2 + \log 5)}{\log 9 - \log 25}$. 2. Use logarithm properties: $\log a - \log b = \log \frac{a}{b}$ and $\log a +

1. **State the problem:** We want to find the quotient of the polynomial division \(\frac{9x^3 + 21x^2 - 18x - 48}{-3x - 6}\). 2. **Simplify the divisor:** Factor out the common fa

1. **State the problem:** Factorise the quadratic expression $$12k^2 - 17k - 7$$. 2. **Identify coefficients:** Here, $$a = 12$$, $$b = -17$$, and $$c = -7$$.

1. The problem is to identify the set of numbers given: 1, 2, 3, 6, -1, -2, -3, -6. 2. These numbers appear to be the divisors of a certain integer because they include both positi

1. **State the problem:** Find the possible real roots of the polynomial $$x^3 - 7x + 6$$ using the Rational Root Theorem. 2. **Recall the Rational Root Theorem:** Possible rationa

1. We are asked to calculate the value of $$\frac{3612 \times 750.9}{113.2 \times 9.98}$$ using logarithms. 2. Recall that logarithms convert multiplication and division into addit

1. Stating the problem: Evaluate the expression $$\arctan \left[ \frac{(1+4 \times 3218)^{1/3}}{\sin (\Re (e^{i \pi}))} \right] / \pi$$

1. The given expression is $12t + 20$. 2. This is a linear algebraic expression in terms of $t$.

1. **State the problem:** Solve the equation $$\frac{4}{3b-3} = \frac{9}{b+3}$$ for $b$. 2. **Rewrite the equation:** We have two fractions set equal: $$\frac{4}{3b-3} = \frac{9}{b