Let's solve the problems step by step.
Problem 1: If \( p = -2 \), \( q = -1 \), and \( r = 3 \), find the value of:
#### 1. \( p^2 + q^2 - r^2 \)
\[
p^2 = (-2)^2 = 4, \quad q^2 = (-1)^2 = 1, \quad r^2 = 3^2 = 9
\]
\[
p^2 + q^2 - r^2 = 4 + 1 - 9 = -4
\]
So, the answer is:
\[
\boxed{-4}
\]
#### 2. \( 2p^2 - q^2 + 3r^2 \)
\[
2p^2 = 2 \cdot 4 = 8, \quad -q^2 = -1, \quad 3r^2 = 3 \cdot 9 = 27
\]
\[
2p^2 - q^2 + 3r^2 = 8 - 1 + 27 = 34
\]
So, the answer is:
\[
\boxed{34}
\]
#### 3. \( p - q - r \)
\[
p - q - r = -2 - (-1) - 3 = -2 + 1 - 3 = -4
\]
So, the answer is:
\[
\boxed{-4}
\]
#### 4. \( p^4 + q^4 - r^4 \)
\[
p^4 = (-2)^4 = 16, \quad q^4 = (-1)^4 = 1, \quad r^4 = 3^4 = 81
\]
\[
p^4 + q^4 - r^4 = 16 + 1 - 81 = -64
\]
So, the answer is:
\[
\boxed{-64}
\]
#### 5. \( p^3 + q^3 + r^3 + 3pqr \)
\[
p^3 = (-2)^3 = -8, \quad q^3 = (-1)^3 = -1, \quad r^3 = 3^3 = 27, \quad 3pqr = 3 \cdot (-2) \cdot (-1) \cdot 3 = 18
\]
\[
p^3 + q^3 + r^3 + 3pqr = -8 - 1 + 27 + 18 = 36
\]
So, the answer is:
\[
\boxed{36}
\]
Problem 6: The side of an equilateral triangle is shown by \( l \). Express the perimeter of the equilateral triangle using \( l \).
An equilateral triangle has all three sides equal. Therefore, the perimeter is:
\[
3l
\]
So, the answer is:
\[
\boxed{3l}
\]
Problem 7: The side of the regular hexagon is denoted by \( l \). Express the perimeter of the hexagon using \( l \).
A regular hexagon has six equal sides. Therefore, the perimeter is:
\[
6l
\]
So, the answer is:
\[
\boxed{6l}
\]
Problem 8: A cube is a three-dimensional figure. It has six faces and all of them are identical squares. The length of an edge of the cube is given by \( l \). Find the formula for the total length of the edges of a cube.
A cube has 12 edges, and each edge has length \( l \). Therefore, the total length of the edges is:
\[
12l
\]
So, the answer is:
\[
\boxed{12l}
\]
Problem 9: Write the constant term of \( 3x^2 + 5x + 8 \).
The constant term in the polynomial \( 3x^2 + 5x + 8 \) is the term that does not contain \( x \). Here, the constant term is:
\[
8
\]
So, the answer is:
\[
\boxed{8}
\]
Problem 10: Write the constant term of \( 2x^2 - 9 \).
The constant term in the polynomial \( 2x^2 - 9 \) is the term that does not contain \( x \). Here, the constant term is:
\[
-9
\]
So, the answer is:
\[
\boxed{-9}
\]
Final Answer:
\[
\boxed{-4, 34, -4, -64, 36, 3l, 6l, 12l, 8, -9}
\]
Parent Tip: Review the logic above to help your child master the concept of 6th grade algebra word problems worksheet.
