🧮 algebra

1. The problem is to simplify the expression $$\frac{1}{\sqrt{n + \sqrt{n^2 - 1}}}$$. 2. Start by letting $$x = \sqrt{n + \sqrt{n^2 - 1}}$$, so the expression becomes $$\frac{1}{x}

1. **State the problem:** Evaluate the sum $$\sum_{n=1}^{12025} \frac{1}{\sqrt{n} + \sqrt{n^2 - 1}}$$

1. **State the problem:** Evaluate the sum $$\sum_{n=1}^{12025} \frac{1}{\sqrt{n} + \sqrt{n^2 - 1}}.$$\n\n2. **Simplify the general term:** Consider the term inside the sum:\n$$a_n

1. **State the problem:** Evaluate the sum

1. The problem is to find the square root of 250. 2. Start by expressing 250 as a product of its prime factors or perfect squares: $$250 = 25 \times 10$$.

1. The problem is to solve the equation $x^2 - y^2 = 3$ for $y$ in terms of $x$. 2. Start by isolating $y^2$ on one side:

1. **State the problem:** Simplify the expression $$\frac{(-5)^3 (-25)^{-1}}{(-5)^{-2}}$$ and write the answer using only positive exponents. 2. **Rewrite the bases:** Note that $$

1. Stating the problem: Simplify the expression $$\frac{3 \frac{1}{3} \times 1 \frac{1}{5}}{3 \frac{1}{3} - 1 \frac{1}{5}}$$. 2. Convert mixed numbers to improper fractions:

1. **State the problem:** Simplify the expression \( \frac{3 \frac{1}{3} \times 1 \frac{1}{5}}{3 \frac{1}{3} - 1 \frac{1}{5}} \). 2. **Convert mixed numbers to improper fractions:*

1. The problem is to evaluate the infinite series $$\sum_{n=1}^\infty (3^{n+1} \cdot 4^{-n}).$$ 2. Rewrite the terms inside the summation to simplify:

1. The problem states that the power of the number three is $n+1$. 2. This means we are dealing with the expression $3^{n+1}$.

1. The problem is to evaluate the expression $+1$ raised to the power of 3. 2. We write the expression as $$+1^3$$.

1. The question asks if the simplified expression is equivalent to the right side of the equation. 2. To verify equivalence, substitute the simplified expression and the right side

1. **State the problem:** Solve the equation $$\ln(e^{2x} + 3) = 2x + \ln 3$$ for $$x$$, correct to 3 decimal places. 2. **Rewrite the equation:** Using properties of logarithms, t

1. **State the problem:** Solve the equation $$\ln(2x - 1) = 2 \ln(x + 1) - \ln x$$ and give the answer correct to 3 decimal places. 2. **Rewrite the right side using logarithm pro

1. Let's clarify the problem you are referring to by restating the step 3 expression or equation. 2. Identify the expressions or operations involved in step 3.

1. **State the problem:** Solve the equation $$\ln(2x - 1) = 2 \ln(x + 1) - \ln x$$ and give the answer correct to 3 decimal places. 2. **Rewrite the right side using logarithm pro

1. **State the problem:** Simplify the expression $$\left(5\omega + 2\omega\right)^2 \times \left(1 - \frac{2}{\omega} + \omega^2\right) \times \left(1 + \omega - \frac{5}{\omega}\

1. **State the problem:** Given the equation $\log_g (2x) - \log_g y = 1$, express $x$ in terms of $y$. 2. **Use logarithm properties:** Recall that $\log_g a - \log_g b = \log_g \

1. Énonçons le problème : vous souhaitez une réponse complète avec toutes les étapes détaillées pour un problème mathématique donné. 2. Comme vous n'avez pas précisé de problème pa

1. Simplify \(\frac{5^x \times 25^{x-1}}{125^{x-1}}\) Step 1: Express all terms with base 5.