Powers and Roots: The perimeter of a square

To calculate the perimeter of a square, we need to first determine the length of one side and then use the formula for the perimeter of a square, which is given by $4 \times \text{side length}$.

The side length is given as $(\sqrt{3}\cdot\sqrt{12}-5)^2\cdot3^2$. Let's simplify this expression step by step:

• First, simplify the square roots: $\sqrt{3}$ and $\sqrt{12}$.

  • $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\times \sqrt{3} = 2 \times \sqrt{3}$
  • Thus, $\sqrt{3} \times \sqrt{12} = \sqrt{3} \times 2 \times \sqrt{3} = 2 \times 3 = 6$.
• Next, substitute back into the expression: $(6 - 5)^2 \cdot 3^2$.
• Simplify inside the parentheses: $6 - 5 = 1$, so we have $1^2 \cdot 3^2$.
• Calculate the squares: $1^2 = 1$ and $3^2 = 9$.
• Multiply the results: $1 \times 9 = 9$.

The side length of the square is therefore $9$.

Thus, the perimeter of the square is:

• $4 \times 9 = 36$.

Therefore, the perimeter of the square is $36$.

To find the perimeter of the square ABCD, we first need to determine the side length of the square using the given area. The area of a square is calculated using the formula: $\text{Area} = s^2$, where $s$ is the side length of the square.

According to the problem, the area of the square is given by the expression $3^2\sqrt{16}$.

Let's simplify this expression:

• First, calculate $3^2$, which is $9$.

• Next, calculate $\sqrt{16}$, which is $4$.

Now, multiply these results: $9 \times 4 = 36$.

Thus, the area of the square is $36$.

Since the area is $s^2 = 36$, we can solve for $s$:

• Find the square root of both sides: $s = \sqrt{36}$.

• This gives $s = 6$.

Now that we have the side length $s = 6$, we can find the perimeter. The perimeter $P$ of a square is given by:

$P = 4s$.

Substituting the side length, we get:

• $P = 4 \times 6 = 24$.

The solution to the question is: $24$

To solve this problem, we need to find the perimeter of a square given its area. The area of the square is given by the expression $(3^2-1^2)+2^3$.

Let us evaluate the expression to find the area:

• Calculate $3^2$, which is $9$.
• Calculate $1^2$, which is $1$.
• Subtract to find $3^2 - 1^2 = 9 - 1 = 8$.
• Calculate $2^3$, which is $8$.
• Add the results: $8 + 8 = 16$.

Therefore, the area of the square is $16$.

In general, the area of a square is given by the formula $s^2$, where $s$ is the side length of the square. To find the side length, we solve the equation:

• $s^2 = 16$.
• Taking the square root of both sides, we find $s = \sqrt{16} = 4$.

The perimeter $P$ of a square with side length $s$ is given by the formula:

• $P = 4s$.

Thus, substituting the value of $s$:

• $P = 4 \times 4 = 16$.

Therefore, the perimeter of the square is $16$.

The problem involves the square ABCD, and we need to determine its perimeter, given the expression for the length of its diagonal. Here's the step-by-step solution:

Let's denote the side of the square ABCD as $s$. The diagonal of a square can be calculated using Pythagoras' theorem as:

• $d = s\sqrt{2}$

The problem provides an expression for the length of the diagonal:

• $3\sqrt{2}\times(3^2-2^3)-2\sqrt{2}$

Let's simplify this expression step by step.

First, calculate the powers:

• $3^2 = 9$

• $2^3 = 8$

Subtract these values:

• $3^2 - 2^3 = 9 - 8 = 1$

Substitute back into the expression for the diagonal:

• $3\sqrt{2} \times 1 - 2\sqrt{2}$

This simplifies to:

• $3\sqrt{2} - 2\sqrt{2}$

• $(3 - 2)\sqrt{2} = 1\sqrt{2} = \sqrt{2}$

So, the length of the diagonal is $\sqrt{2}$.

We know from the formula for the diagonal of a square that $d = s\sqrt{2}$. Given $d = \sqrt{2}$, we can equate:

• $s\sqrt{2} = \sqrt{2}$

Thus:

• $s = 1$

Therefore, the perimeter of the square ABCD is:

• $4 \times s = 4 \times 1 = 4$

Hence, the perimeter of the square ABCD is 4.