🧮 algebra

1. The problem is to simplify the expression consisting of two fractions: $\frac{a^3}{1}$ and $\frac{a^3}{4}$.\n\n2. Write the expression clearly: $$\left[ \frac{a^3}{1}, \frac{a^3

1. **State the problem:** There were some children at a party. Each child took a gift for each of their friends, and altogether there were 81 gifts. We need to find out how many ch

1. **State the problem:** Simplify the expression $$(a^{7})^{4} (a^{5})^{1}$$ and identify the simplest form. 2. Use the power of a power rule: $$(a^{m})^{n} = a^{m \cdot n}$$.

1. State the problem: We need to simplify the expression $$(5r + 7)(5r - 7)$$. 2. Recognize the form: This is a product of two binomials in the form $(a + b)(a - b)$, which is a di

1. Stated problem: Multiply the binomials $ (4n + 3)(n + 9) $. 2. Use the distributive property (FOIL method) to multiply each term in the first binomial by each term in the second

1. **Stating the problem:** Simplify and solve the equation $$31 \times 7(t^2 + 5t - 9) + t = t(7t - 2) + 13$$ for $t$. 2. **Expand the left side:**

1. Problem a: Simplify $a^{3/2} \cdot a^{4/3}$. Use the exponent multiplication rule $a^m \cdot a^n = a^{m+n}$:

1. **State the problem:** Simplify the algebraic expression $$2ab(7a^4b^2 + a^5b - 2a)$$. 2. **Distribute** $2ab$ to each term inside the parentheses:

1. **State the problem:** Solve the system of equations \(8 = 2^{x+y}\) and \(1 = 3^{x-y}\) simultaneously. 2. **Rewrite the equations using properties of exponents:**

1. **State the problem:** We want to solve the system of equations simultaneously: $$8 = 2(x+y)$$

1. **Find the domain and range of the function** $h(x) = \sqrt{x^2 - 4}$. Step 1: The expression under the square root, called the radicand, must be non-negative for $h(x)$ to be r

1. The problem is to determine the value or nature of "H", which is not clearly specified. 2. Since no further context or equation involving "H" is given, it is impossible to proce

1. The domain of a function is the set of all possible input values (usually $x$ values) for which the function is defined. 2. To determine the domain, identify values of $x$ that

1. We are asked to express the relationship between time spent on a job and the fee charged by a carpenter. 2. Let $t$ be the time in hours and $F(t)$ be the fee charged in euros.

1. Stating the problem: We have the function $$f(x)=\frac{2x-1}{x-1}$$ and we want to express it as a division of two functions $$u(x)$$ and $$v(x)$$ such that $$f(x) = \frac{u(x)}

1. The problem is to divide the function $$f(x)=\frac{2x-1}{x-1}$$ into two functions $$u(x)$$ and $$v(x)$$ such that $$f(x) = \frac{u(x)}{v(x)}$$. 2. Observe the given function $$

1. **Problem Statement:** Factorize the given polynomials. 2. **viii.** Factorize $-6x^2 - x + 2$.

1. Stating the problem: Solve the linear equation $1x+\frac{130}{180}-3x=\frac{31}{27}$.\n\n2. Simplify the fractions where possible:\n$\frac{130}{180} = \frac{13}{18}$.\nSo equati

1. The problem is to identify and analyze the graph of a function described with points and behavior. 2. The graph passes through the point (1, 2). This means that the function's v

1. The problem is to find the function describing the situation. 2. Since no specific problem details were given, I will explain how to approach defining a function from a problem

1. The problem involves analyzing a curve that passes through points and describes a specific behavior between those points. 2. Key points given are approximately at $x=-1$, $y\les