🧮 algebra

1. State the problem: Solve the equation $12^x - 2 = 3^{3x} \cdot 2^{6x}$ for $x$.\n\n2. Express bases with prime factorization: $12 = 2^2 \cdot 3$, so $12^x = (2^2 \cdot 3)^x = 2^

1. Simplify $5y^{2} \times 4y^{4}$: Multiply coefficients: $5 \times 4 = 20$

1. You asked to graph and solve an equation, but you did not specify the equation itself. 2. Please provide the exact equation or function you want to solve and plot so I can assis

1. **State the problem:** We have 12 oranges that need to be divided into bags. Each bag must have the same number of oranges.

1. We are given the equation $$\frac{8}{5} = \frac{?}{20}$$ and need to find the missing numerator. 2. To solve for the missing number, we use the property of equivalent fractions

1. The problem provides a system of linear equations: $$4x - 2y = 24$$

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.