Question

9.) simplify the expression below using order of operations.
\\ -9^2 - (-6) \times 2 + 8^2 \\

10.) simplify the expression below using order of operations.
\\ -2 - \frac{(-2)^3}{12^2 - 9} \\

11.) the area of the triangle below is \\(\frac{12}{25}\\) square feet. what is the length of the base? express your answer as a fraction in simplest form.

12.) what is the area, in square inches, of the shape?

13.) given the two rectangles below. find the area of the shaded region.

Explanation:

Evaluate Question 9

Evaluate the expression using the order of operations:

$$ -9^2 - (-6) \times 2 + 8^2 $$

First, evaluate the exponents:

$$ -9^2 = -81 $$

$$ 8^2 = 64 $$

Substitute these back:

$$ -81 - (-6) \times 2 + 64 $$

Next, perform the multiplication:

$$ -6 \times 2 = -12 $$

Substitute back:

$$ -81 - (-12) + 64 = -81 + 12 + 64 $$

Finally, perform addition and subtraction from left to right:

$$ -81 + 12 = -69 $$

$$ -69 + 64 = -5 $$

Evaluate Question 10

Evaluate the expression using the order of operations:

$$ -2 - \frac{(-2)^3}{12^2 - 9} $$

First, evaluate the numerator exponent:

$$ (-2)^3 = -8 $$

Next, evaluate the denominator exponent:

$$ 12^2 = 144 $$

Simplify the denominator:

$$ 144 - 9 = 135 $$

Substitute these values back into the fraction:

$$ -2 - \frac{-8}{135} = -2 + \frac{8}{135} $$

Convert \(-2\) to a fraction with a denominator of 135:

$$ -2 = -\frac{270}{135} $$

Combine the fractions:

$$ -\frac{270}{135} + \frac{8}{135} = -\frac{262}{135} $$

Evaluate Question 11

Using the Area of Triangles formula:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Given:

$$ \text{Area} = \frac{12}{25}\text{ sq ft},\quad \text{height} = \frac{4}{5}\text{ ft} $$

Substitute the values into the formula:

$$ \frac{12}{25} = \frac{1}{2} \times \text{base} \times \frac{4}{5} $$

Using Fraction Multiplication:

$$ \frac{12}{25} = \text{base} \times \frac{2}{5} $$

Isolate the base using Fraction Division:

$$ \text{base} = \frac{12}{25} \div \frac{2}{5} = \frac{12}{25} \times \frac{5}{2} = \frac{6}{5}\text{ ft} $$

Evaluate Question 12

Find the area of the right triangle:

$$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$

Given:

$$ \text{base} = 9\text{ in},\quad \text{height} = 2.3\text{ in} $$

Calculate the area:

$$ \text{Area} = \frac{1}{2} \times 9 \times 2.3 = 4.5 \times 2.3 = 10.35\text{ sq in} $$

Evaluate Question 13

Find the area of the shaded region.
The outer region is a large rectangle.
The inner region is a smaller rectangle.
First, find the dimensions of the inner rectangle:

$$ \text{Width} = 9\text{ units},\quad \text{Height} = 6\text{ units} $$

$$ \text{Area}_{\text{inner}} = 9 \times 6 = 54 $$

Next, find the dimensions of the outer rectangle:

$$ \text{Width} = 9 + 2 = 11\text{ units} $$

$$ \text{Height} = 6 + 2 = 8\text{ units} $$

$$ \text{Area}_{\text{outer}} = 11 \times 8 = 88 $$

Subtract the inner area from the outer area to find the shaded area:

$$ \text{Area}_{\text{shaded}} = 88 - 54 = 34 $$

Answer:

Question 9

\(-5\)

Question 10

\(-\frac{262}{135}\)

Question 11

\(\frac{6}{5}\)

Question 12

\(10.35\)

Question 13

\(34\)