🧮 algebra

Problem: Fill the missing exponent $x$ in each expression so that the left number raised to $x$ equals the right number. 1. Set up the equation $$2^{x}=81$$.

1. Solve $|m| = 42$. Since absolute value equals 42, $m = 42$ or $m = -42$.

1. The problem involves finding the missing numbers (represented by boxes) in sequences where numbers and their powers or roots are related. 2. For the first sequence: 2 ☐ 3 ☐ 4 ☐

1. དྲི་བ་གསལ་བའི་མཚན་རྟགས་གི་དོན་ལུ་ $2x+5=13$ ཞེས་འདི་ཡིན། 2. དེ་ལས་ $x$ ག་རེ་ཡིན་ན་ཞེས་འདི་གནང་བ་ཡིན།

1. དྲི་བ་གསལ་བའི་མཚན་རྟགས་གི་དོན་ལུ་གསལ་བའི་གསལ་བ་གནང་བ་ཡིན། 2x + 5 = 13 ནི x ག་རེ་ཡིན་ན་ཞིག་གི་དྲི་བ་ཡིན། 2. དེ་ལས་ x ག་རེ་ཡིན་ན་ཞིག་གི་དོན་ལུ་གསལ་བའི་གསལ་བ་གནང་བ་ཡིན། དེ་ལས་ 2x +

1. Find the inverse of $f(x) = 7 - 3x$. Step 1: Replace $f(x)$ with $y$:

1. **State the problem:** Simplify the expression $$\frac{-216 \left(e^{\frac{1}{2}}\right)^5}{\left(3 e^{\frac{1}{4}}\right)^2}$$

1. The problem is to simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. First, factor the numerator: $$2x^2 - 8 = 2(x^2 - 4)$$.

1. Diberikan bahwa harga sebuah buku latihan adalah dua kali harga sebuah air minuman. 2. Misalkan harga air minuman adalah $x$.

1. **State the problem:** Find the local maximum and minimum points of the function $$f(x) = x^3 + 3x^2 - 4x$$ given its roots at $$x = -4, 0, 1$$. 2. **Find the derivative:** To f

1. The problem is to find the equation of the line passing through the points $(-5,0)$ and $(0,4)$ using the slope formula. 2. The slope $m$ is given by the formula:

1. Simplify each radical expression by factoring out perfect squares and applying the property $\sqrt{a^2} = a$. 2. For example, for problem 15: $\sqrt{36x^2y^3} = \sqrt{36} \cdot

1. Problem: Simplify $\sqrt{125n}$. - Factor inside the radical: $125 = 25 \times 5$.

1. **State the problem:** Simplify the expression $$\frac{x^2 + 5x + 6}{x^2 + 2x}$$. 2. **Factor the numerator:** The quadratic $x^2 + 5x + 6$ factors into $(x + 2)(x + 3)$ because

1. Simplify $\sqrt{75}$: $75 = 25 \times 3$, so $\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$.

1. **State the problem:** We have a 100 gram metal alloy that is 40% silver. It is made by mixing two metal alloys: the first is 65% silver and the second is 25% silver. We need to

1. Problem 10: Find the y-intercept of a line with slope $\frac{3}{4}$ passing through the point $(6, -2)$. 2. Use the point-slope form of a line: $$y - y_1 = m(x - x_1)$$ where $m

1. **Find the slope of the line passing through the points** $\left(-\frac{5}{2},4\right)$ and $\left(\frac{3}{2}, -2\right)$. The slope formula is

1. The problem is to graph the linear function $$y = -\frac{1}{4}x + \frac{1}{2}$$ by plotting its intercepts. 2. To find the y-intercept, set $$x = 0$$:

1. Stating the problem: Simplify the expression $$\frac{n \times n + n}{n} = 10$$ and solve for $n$. 2. Simplify the numerator: $$n \times n + n = n^2 + n$$.

1. **State the problem:** We want to analyze the function $$f(x) = \frac{x^{\frac{3}{2}} + 1}{x^{\frac{1}{2}} - x}$$ and simplify or understand its behavior. 2. **Rewrite the funct