🧮 algebra

1. **Problem:** Given $A = \sqrt{5} + 2$ and $B = \sqrt{5} - 2$, calculate $A^2$, $B^2$, and $AB$. - Calculate $A^2$:

**Problem:** Given $A = \sqrt{5} + 2$ and $B = \sqrt{5} - 2$, find $A^2$, $B^2$, $A \times B$, and verify that $\frac{A}{B} + \frac{B}{A}$ is a number. 1. Calculate $A^2$:

1. **State the problem:** Simplify the expression $Y + xy + x + xy$. 2. **Group like terms:** Notice that $xy$ appears twice, so we can write the expression as $Y + x + xy + xy$.

1. The problem is to solve for $x$ in the equation $4^x = 12$. 2. We take the natural logarithm (ln) on both sides to handle the exponent: $$\ln(4^x) = \ln(12)$$

1. **Problem statement:** Simplify the expression $$\frac{2^{-2} + 5 \times \left(\frac{1}{3}\right)^0 \times \left(\frac{4}{5}\right)^{-1}}{3 - \left(\frac{2}{3}\right)^{-2} \time

1. The problem is to evaluate $\log 8$ with the default assumption of base 10. 2. Recall that $\log 8 = \log_{10} 8$ means the power to which 10 must be raised to get 8.

1. The problem is to evaluate $\log 3$, which usually means the logarithm of 3 in base 10 if no base is specified. 2. Recall the definition: $\log_b a$ is the exponent to which we

1. Since the request mentions tasks in pictures but no pictures are provided, please upload the images or describe the tasks for assistance. 2. I will then help solve the math prob

1. The problem is to solve the equation $10^x = 5$ for $x$. 2. To solve for $x$, take the logarithm base 10 of both sides: $$\log_{10}(10^x) = \log_{10}(5)$$

1. The given expression is \left(\;\right)\left(\;\right), which represents the multiplication of two empty parentheses without any visible terms inside. 2. Since there are no term

1. We are asked to compute the sum of the numbers $53 \times 10^{-4} + 32 \times 10^{-3} - 16 \times 10^{-5}$. 2. Convert all terms to the same power of ten for easier addition. Le

1. **State the problem:** We have total tickets sold = 100 and total money collected = 590. 2. Let $a$ be the number of adult tickets and $c$ be the number of child tickets. So, we

1. Simplify each expression by factoring and reducing common terms. (a) Simplify $$\frac{6m - 8n}{7pq - 7pr} \times \frac{9qs + 9rs}{3m^2 - 4mn}$$

1. Stating the problem: We are given ratios $A:B=2:3$, $B:C=4:5$, $C:D=7:10$ and asked to find the values of $A$, $B$, $C$, and $D$. 2. Since the problem involves direct and invers

1. State the problem: Expand the logarithmic expression $$\log_3 \left[ \frac{18 (x + 2)^2}{(x - 2)^3 (x + 5)^2} \right]$$ using laws of logarithms. 2. Recall the logarithm laws us

1. **Problem 1:** Solve the equation \((2 + x)(5 + 2x) = 0\). 2. To solve, we use the zero product property: if \(AB = 0\), then either \(A = 0\) or \(B = 0\).

1. We are asked to convert the repeating decimal $0.13\overline{5}$ into a fraction in its simplest form. 2. Let $x = 0.1355555\dots$ where the digit $5$ repeats indefinitely.

1. The user mentioned "In the previous equation" but did not provide any equation or problem statement. 2. To assist effectively, please provide the specific equation or problem yo

1. We are given the polynomial expression $x^4 - 4x^3 + 10$ and need to analyze or simplify it as needed. 2. First, observe that this is a quartic polynomial with terms $x^4$, $-4x

1. **Problem**: Find the zeros of the quadratic polynomial $P(x) = x^2 + 7x + 10$ and verify the relationship between the zeros and coefficients. 2. **Identify the polynomial**: Gi

1. The problem is to find the zero of the polynomial $p(x) = x^2 + 7 \times 10$ and verify the relationship between its zero and coefficients. 2. First, simplify the polynomial: