PIL import Image import pytesseract

Load the uploaded image to extract text

image_path = "/mnt/data/17332458326278nyppohx.jpg" extracted_text = pytesseract.image_to_string(Image.open(image_path))

extracted_textHere are the questions extracted from the image:

1. The volume of a cube is given as $8.1 \times 10^{18} \, \text{cm}^3$. Find the side length of the cube, expressed in standard form.

2. The area of a square is $2.56 \times 10^{14} \, \text{m}^2$. Find the side length of the square, expressed in standard form.

3. Simplify: $\sqrt[3]{1.728 \times 10^{-12}}$.

Solutions:

1. Side length of the cube:

The formula for the volume of a cube is: $V = a^3$ where $a$ is the side length. Rearranging: $a = \sqrt[3]{V}$

Substitute $V = 8.1 \times 10^{18}$: [ a = \sqrt[3]{8.1} \times \sqrt[3]{10^{18}} ]

$\sqrt[3]{8.1} \approx 2$, and $\sqrt[3]{10^{18}} = 10^6$.

$a = 2 \times 10^6 = 2.0 \times 10^6 \, \text{cm}.$

2. Side length of the square:

The formula for the area of a square is: $A = a^2$ where $a$ is the side length. Rearranging: $a = \sqrt{A}$

Substitute $A = 2.56 \times 10^{14}$: $a = \sqrt{2.56} \times \sqrt{10^{14}}$

$\sqrt{2.56} = 1.6$, and $\sqrt{10^{14}} = 10^7$.

$a = 1.6 \times 10^7 \, \text{m}.$

3. Simplify $\sqrt[3]{1.728 \times 10^{-12}}$:

[ \sqrt[3]{1.728 \times 10^{-12}} = \sqrt[3]{1.728} \times \sqrt[3]{10^{-12}} ]

$\sqrt[3]{1.728} = 1.2$, and $\sqrt[3]{10^{-12}} = 10^{-4}$.

$\sqrt[3]{1.728 \times 10^{-12}} = 1.2 \times 10^{-4}.$

Final Answers:

1. $2.0 \times 10^6 \, \text{cm}$
2. $1.6 \times 10^7 \, \text{m}$
3. $1.2 \times 10^{-4}$

Would you like further explanation or have questions?

Related Questions:

1. What is the formula for finding the volume of other geometric shapes?
2. How do you calculate cube roots manually or using a calculator?
3. Can you describe the properties of standard form in mathematics?
4. How would the answer change if the area of the square was $2.25 \times 10^{12}$?
5. What is the significance of exponents in scientific notation?

Tip: When simplifying scientific notation, handle the coefficients and powers of ten separately to avoid confusion.