Find The Value Of M M M For Which 2 M ÷ 2 − 3 = 29 2^m \div 2^{-3} = 29 2 M ÷ 2 − 3 = 29 .10. Subtract: 3 X Y + 5 Y Z − 7 Z X 3xy + 5yz - 7zx 3 X Y + 5 Yz − 7 Z X From 5 X Y − 2 Y Z − 2 Z X + 10 X Y Z 5xy - 2yz - 2zx + 10xyz 5 X Y − 2 Yz − 2 Z X + 10 X Yz .11. Factorize: 81 X 2 − 25 81x^2 - 25 81 X 2 − 25 .

Section 1: Solving Exponential Equations

9. Find the value of $m$ for which $2^m \div 2^{-3} = 29$

When dealing with exponential equations, we need to apply the rules of exponents to simplify the expression and solve for the variable. In this case, we have the equation $2^m \div 2^{-3} = 29$. To simplify this expression, we can use the rule that states $\frac{a^m}{a^n} = a^{m-n}$.

Using this rule, we can rewrite the equation as $2^m \div 2^{-3} = 2^{m-(-3)} = 2^{m+3}$. Now, we can set this expression equal to 29 and solve for $m$.

$2^{m+3} = 29$

To solve for $m$, we can take the logarithm of both sides of the equation. We can use any base for the logarithm, but let's use the natural logarithm (base $e$).

$\ln(2^{m+3}) = \ln(29)$

Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can rewrite the equation as:

$(m+3)\ln(2) = \ln(29)$

Now, we can solve for $m$ by dividing both sides of the equation by $\ln(2)$.

$m+3 = \frac{\ln(29)}{\ln(2)}$

$m = \frac{\ln(29)}{\ln(2)} - 3$

Using a calculator to evaluate the expression, we get:

$m \approx 5.04 - 3$

$m \approx 2.04$

Therefore, the value of $m$ for which $2^m \div 2^{-3} = 29$ is approximately 2.04.

10. Subtract: $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$

To subtract one expression from another, we need to combine like terms and simplify the resulting expression. In this case, we have the expression $3xy + 5yz - 7zx$ that we need to subtract from the expression $5xy - 2yz - 2zx + 10xyz$.

First, let's rewrite the expression $3xy + 5yz - 7zx$ as a single expression by combining like terms.

$3xy + 5yz - 7zx = (3xy) + (5yz) + (-7zx)$

Now, we can subtract this expression from the expression $5xy - 2yz - 2zx + 10xyz$.

$(5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx)$

To subtract the expressions, we need to combine like terms. We can do this by adding or subtracting the coefficients of the like terms.

$5xy - 3xy = 2xy$

$-2yz - 5yz = -7yz$

$-2zx + 7zx = 5zx$

$10xyz$ remains the same.

The resulting expression is:

$2xy - 7yz + 5zx + 10xyz$

Therefore, the result of subtracting $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$ is $2xy - 7yz + 5zx + 10xyz$.

11. Factorize: $81x^2 - 25$

To factorize an expression, we need to find two binomials whose product is equal to the given expression. In this case, we have the expression $81x^2 - 25$ that we need to factorize.

First, let's recognize that the expression $81x^2 - 25$ is a difference of squares. We can rewrite it as:

$(9x)^2 - 5^2$

Now, we can use the formula for the difference of squares, which is:

$a^2 - b^2 = (a + b)(a - b)$

Using this formula, we can factorize the expression as:

$(9x + 5)(9x - 5)$

Therefore, the factorized form of the expression $81x^2 - 25$ is $(9x + 5)(9x - 5)$.

Section 2: Solving Algebraic Expressions

9. Find the value of $x$ for which $3x + 5 = 11$

To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 5 from both sides of the equation.

$3x + 5 - 5 = 11 - 5$

$3x = 6$

Now, we can divide both sides of the equation by 3 to solve for $x$.

$x = \frac{6}{3}$

$x = 2$

Therefore, the value of $x$ for which $3x + 5 = 11$ is 2.

10. Simplify: $\frac{4x^2 + 12x + 9}{2x + 3}$

To simplify the expression, we need to factorize the numerator and cancel out any common factors.

First, let's factorize the numerator $4x^2 + 12x + 9$.

$4x^2 + 12x + 9 = (2x + 3)(2x + 3)$

Now, we can rewrite the expression as:

$\frac{(2x + 3)(2x + 3)}{2x + 3}$

We can cancel out the common factor $(2x + 3)$ from the numerator and denominator.

$\frac{(2x + 3)(2x + 3)}{2x + 3} = 2x + 3$

Therefore, the simplified form of the expression $\frac{4x^2 + 12x + 9}{2x + 3}$ is $2x + 3$.

11. Solve: $x^2 + 4x + 4 = 0$

To solve the quadratic equation, we need to factorize the left-hand side of the equation.

$x^2 + 4x + 4 = (x + 2)(x + 2)$

Now, we can rewrite the equation as:

$(x + 2)(x + 2) = 0$

We can set each factor equal to 0 and solve for $x$.

$x + 2 = 0$

$x = -2$

Therefore, the solution to the quadratic equation $x^2 + 4x + 4 = 0$ is $x = -2$.

Conclusion

Section 1: Algebra

Q1: What is the value of $x$ in the equation $2x + 5 = 11$?

A1: To solve for $x$, we need to isolate the variable $x$ on one side of the equation. We can do this by subtracting 5 from both sides of the equation.

$2x + 5 - 5 = 11 - 5$

$2x = 6$

Now, we can divide both sides of the equation by 2 to solve for $x$.

$x = \frac{6}{2}$

$x = 3$

Therefore, the value of $x$ in the equation $2x + 5 = 11$ is 3.

Q2: Simplify the expression $\frac{3x^2 + 6x + 9}{3x + 3}$.

A2: To simplify the expression, we need to factorize the numerator and cancel out any common factors.

First, let's factorize the numerator $3x^2 + 6x + 9$.

$3x^2 + 6x + 9 = (3x + 3)(x + 3)$

Now, we can rewrite the expression as:

$\frac{(3x + 3)(x + 3)}{3x + 3}$

We can cancel out the common factor $(3x + 3)$ from the numerator and denominator.

$\frac{(3x + 3)(x + 3)}{3x + 3} = x + 3$

Therefore, the simplified form of the expression $\frac{3x^2 + 6x + 9}{3x + 3}$ is $x + 3$.

Q3: Solve the quadratic equation $x^2 + 5x + 6 = 0$.

A3: To solve the quadratic equation, we need to factorize the left-hand side of the equation.

$x^2 + 5x + 6 = (x + 2)(x + 3)$

Now, we can rewrite the equation as:

$(x + 2)(x + 3) = 0$

We can set each factor equal to 0 and solve for $x$.

$x + 2 = 0$

$x = -2$

$x + 3 = 0$

$x = -3$

Therefore, the solutions to the quadratic equation $x^2 + 5x + 6 = 0$ are $x = -2$ and $x = -3$.

Section 2: Geometry

Q1: What is the perimeter of a rectangle with length 6 and width 4?

A1: To find the perimeter of a rectangle, we need to add up the lengths of all four sides.

Perimeter = 2(length + width)

Perimeter = 2(6 + 4)

Perimeter = 2(10)

Perimeter = 20

Therefore, the perimeter of the rectangle is 20.

Q2: What is the area of a triangle with base 5 and height 6?

A2: To find the area of a triangle, we need to use the formula:

Area = (base × height) / 2

Area = (5 × 6) / 2

Area = 30 / 2

Area = 15

Therefore, the area of the triangle is 15.

Q3: What is the volume of a cube with side length 4?

A3: To find the volume of a cube, we need to cube the side length.

Volume = side length^3

Volume = 4^3

Volume = 64

Therefore, the volume of the cube is 64.

Section 3: Trigonometry

Q1: What is the value of sin(30°)?

A1: To find the value of sin(30°), we need to use the unit circle.

sin(30°) = 0.5

Therefore, the value of sin(30°) is 0.5.

Q2: What is the value of cos(60°)?

A2: To find the value of cos(60°), we need to use the unit circle.

cos(60°) = 0.5

Therefore, the value of cos(60°) is 0.5.

Q3: What is the value of tan(45°)?

A3: To find the value of tan(45°), we need to use the unit circle.

tan(45°) = 1

Therefore, the value of tan(45°) is 1.

Conclusion

In this article, we have answered several questions on various mathematical topics, including algebra, geometry, and trigonometry. We have provided step-by-step solutions to each problem, making it easy for readers to follow along and understand the concepts.