🧮 algebra

1. \text{Рассмотрим уравнение: } x^2 - 5x + 6 = 0. 2. \text{Факторизуем } x^2 - 5x + 6 \text{ как } (x - 2)(x - 3) = 0.

1. **State the problem:** Given the ratios $\frac{a}{b} = \frac{3}{5}$ and $\frac{b}{c} = \frac{4}{5}$, find the ratio $a : b : c$ in simplest form.

1. The problem involves understanding the use of equations like $P(x) \rightarrow Q(x)$, which typically denotes that $P(x)$ implies $Q(x)$ or a transformation from one function or

1. The problem asks us to write a formula for the total number of doughnuts Tom buys, denoted by $T$, given he buys $B$ bags and $C$ cartons of doughnuts. 2. To express $T$ in term

1. **State the problem:** Given the system of equations: $$a^7 + b^7 = n$$

1. **State the problem:** We need to find the value of $x$ in the equation $$\frac{(5^{12})^4}{5^8} = 5^x.$$ \n\n2. **Simplify the numerator:** Using the power of a power rule, $$

1. The problem states that the original price of the necklace was £54 and it is reduced by 8.5% in the sale. 2. To find the sale price, we first calculate the amount of discount us

1. We start with the given equation: $$(8^3 \times 8^4)^5 = 8^w$$ 2. Use the property of exponents for multiplication: $$a^m \times a^n = a^{m+n}$$ to combine $8^3$ and $8^4$. So,

1. The problem asks which two expressions are equivalent to $4^{-3}$. 2. Recall that negative exponents mean the reciprocal: $4^{-3} = \frac{1}{4^3}$.

1. **State the problem:** Solve the equation $$\left(\frac{3}{4}\right)^{2x-1} \cdot \left(\frac{4}{3}\right)^{x+2} = \frac{9}{16}.$$\n\n2. **Rewrite the bases to facilitate simpli

1. The problem is to evaluate the expression $1000^{-\frac{3}{2}}$. 2. Recall that a negative exponent means we take the reciprocal:

1. The problem asks to find the value of $$1000^{\left(-\frac{2}{3}\right)}$$. 2. Start by rewriting the base number 1000 as a power of 10: $$1000 = 10^3$$.

1. State the problem: Solve for $d$ in the equation $$(17^3)^d = 17^{12}$$. 2. Use the power of a power rule: When raising an exponential expression to another power, multiply the

1. **State the problem:** We need to find the difference between the sale prices of two houses. Each house has an original price and a percentage discount offer. We will calculate

1. The problem is to match the equation $y=2^x$ with its correct graph. 2. The equation $y=2^x$ represents an exponential function with base 2.

1. The problem asks to match the equation $y=5x$ with its corresponding graph. 2. The equation $y=5x$ is a linear equation representing a straight line passing through the origin.

1. We are asked to simplify or analyze the function $$y=\frac{2x^4 \tan x}{e^{2x} \sin x}$$. 2. Recall that $$\tan x = \frac{\sin x}{\cos x}$$, so substitute this in to rewrite the

1. We are given the function $$y = x(x - 1)$$ and the domain $$\{-2, -1, 0, 1, 2\}$$. 2. To find the range, substitute each value from the domain into the function and calculate th

1. We are asked to find the value of $h$ in the equation $$(19^3)^{-8} = 19^h.$$ 2. Recall the exponent rule: $$(a^m)^n = a^{m \cdot n}.$$

1. **State the problem:** We need to find the value of $f$ in the equation $$15^4 - 7 = 15^f.$$\n\n2. **Evaluate the left side expression:** Calculate $15^4$.\n$$15^4 = 15 \times 1

1. The problem asks to find the value of the function $f(x)$ at $x = -2.3$ given the function $f(x) = 12$. 2. Since $f(x) = 12$ is a constant function, it means that for every inpu