🧮 algebra

**Problem:** Simplify the expression $$\sqrt{x}^{\frac{1}{2}} \cdot (yz)^{-\frac{1}{2}} \cdot \sqrt{z}^3 \cdot y^{\frac{1}{2}} \cdot (x^{\frac{1}{4}})^3$$. 1. Rewrite all radicals

1. The problem is to simplify the expression $t^2 + \frac{2}{t^2} + 0 \times t + 1$. 2. Notice that $0 \times t = 0$, so it can be removed from the expression.

1. State the problem: Simplify the expression $t \times 2$. 2. Understand that multiplying a variable by a number means doubling that variable.

1. The problem states that the ratio (common ratio) of a geometric progression (G.P.) is given as \(\frac{25}{49}\). We have the second term as \(x + 5\). 2. Let's denote the first

1. The problem presents rows of numbers with two unknown values labeled as $X$ and $Y$. Our goal is to find a relationship or rule to determine $X$ and $Y$ for each row. 2. Let's a

1. Please provide the full problem or expression you'd like me to calculate so I can assist you accurately.

1. The problem involves calculating the price per year given two variables: Cost per kg and Price per year as indicated in the last column. 2. The equation based on the problem sta

1. The problem asks to calculate the price per year for each raw material by multiplying the cost per kg by the amount per year (in kg). 2. The formula to use is:

1. The problem is to find the prime factorization of the numbers represented in the diagram: 2, 5, 45, 1, 3, and 15. 2. Prime factorization means expressing each number as a produc

1. The problem is to compute the value of $22^5$. 2. Recall that $22^5$ means $22$ multiplied by itself $5$ times: $$22^5 = 22 \times 22 \times 22 \times 22 \times 22$$.

1) **Determine whether the sequence converges or diverges, and if it converges, find its limit.** **a)** $a_n = \frac{3 + 5n^2}{n + n^2}$

1. State the problem: Simplify the expression $$ (2a + 2b)^2 + (2a - 2b)^2 $$. 2. Expand each square term using the formula $$(x + y)^2 = x^2 + 2xy + y^2$$ and $$(x - y)^2 = x^2 -

1. State the problem: Simplify $|x|^3$. 2. Interpretation: I read this as the cube of the absolute value, which is $|x|^3$.

1. Stating the problem: Calculate the total cost for each raw material given the cost per kg and amount used per year, then find the total cost of all materials combined. 2. Calcul

1. Let's start by defining the problem: Polynomial division is a method to divide one polynomial (the dividend) by another polynomial (the divisor), similar to long division with n

1. **State the problem:** Solve the equation $-x + 1 = (x - 1)^2 - 2$. 2. **Expand the right side:**

1. **Statement of Problem**: Given two propositions $P$ and $Q$: - $(P): \exists x \in \mathbb{R} : x^2 + x - 2 = 0$

1. The original problem had \( \sqrt{25} \), which equals 5. 2. Now we replace \( \sqrt{25} \) with \( \sqrt{15} \).

1. **Problem 2:** Given functions $f(x) = a + x^2$ and $L(x) = c$ where $a$, $c$ are constants, and the equation $3f(2) + 3L(x) = 6$ holds for all $x$. We need to find $2f(0) + 2L(

1. The problem is to multiply two binomials using the extended method (also known as the FOIL method). Let's consider the binomials $(a+b)$ and $(c+d)$. 2. The extended method mult

1. **Problem 9 (a):** Find the natural domain and range of $f(x)=\frac{1}{x-3}$. Domain: The denominator cannot be zero, so $x-3\neq0 \Rightarrow x\neq3$.