Abbreviated Multiplication Formulas Practice Problems

To solve this problem, we'll follow these steps:

• Step 1: Identify a and b in the expression $(x^2 - 6)^2$.
• Step 2: Apply the square of a difference formula.
• Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: The expression is $(x^2 - 6)^2$. Here, $a = x^2$ and $b = 6$.
Step 2: Apply the binomial formula: $(a - b)^2 = a^2 - 2ab + b^2$.
Step 3:
1. Calculate $a^2$:
$a^2 = (x^2)^2 = x^4$.
2. Calculate $2ab$:
$2ab = 2(x^2)(6) = 12x^2$.
3. Calculate $b^2$:
$b^2 = 6^2 = 36$.
4. Substitute these back into the formula:
$(x^2 - 6)^2 = x^4 - 12x^2 + 36$.

Therefore, the expanded expression is $x^4 - 12x^2 + 36$.

To solve this problem, let's start by identifying the parts of the binomial:

• The expression $(3x-y)^2$ represents a binomial squared.
• We recognize it has the form $(a-b)^2$ where $a = 3x$ and $b = y$.
• Using the formula for the square of a difference: $(a-b)^2 = a^2 - 2ab + b^2$, we find the expanded form.

Let's apply the formula:

Step 1: Expand $(3x-y)^2$ using the formula:
$(3x-y)^2 = (3x)^2 - 2(3x)(y) + y^2$

Step 2: Calculate each part:
$(3x)^2 = 9x^2$
$-2(3x)(y) = -6xy$
$y^2$ stays as $y^2$

Step 3: Combine these results to get the addition form:
$9x^2 - 6xy + y^2$

The expression in multiplication form, as provided, is just repeating the factors:
$(3x-y)(3x-y)$

Therefore, the expression rewritten as addition is $9x^2 - 6xy + y^2$ and as multiplication $(3x-y)(3x-y)$.

To solve this problem, we will apply the square of a binomial formula.

The given expression is $(4b-3)(4b-3)$. We recognize this as the square of a binomial, which can be rewritten as $(4b-3)^2$. To expand this expression, we use the formula:

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

In our expression, $a = 4b$ and $b = 3$. Let's apply the formula:

• Calculate $a^2$:
  $a^2 = (4b)^2 = 16b^2$
• Calculate $-2ab$:
  $-2ab = -2(4b)(3) = -24b$
• Calculate $b^2$:
  $b^2 = (3)^2 = 9$

Putting it all together, we have:

$(4b-3)^2 = 16b^2 - 24b + 9$

Therefore, the exponential summation expression is $(4b-3)^2$, with the expanded form:

$16b^2 - 24b + 9$

This matches choice 3, confirming our solution.

To solve for $(7b - 3x)^2$ as a sum, we'll follow these steps:

• Step 1: Identify the given expression and apply the formula for the square of a difference:
  $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 7b$ and $b = 3x$.
• Step 2: Expand each term:
• $a^2 = (7b)^2 = 49b^2$
• $-2ab = -2 \times 7b \times 3x = -42bx$
• $b^2 = (3x)^2 = 9x^2$
• Step 3: Combine all terms to form the sum:
  $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Therefore, the solution to the problem is $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Hence, the correct answer choice is: $49b^2 - 42bx + 9x^2$

To solve the expression $(x-x^2)^2$, we will use the square of a binomial formula $(a-b)^2 = a^2 - 2ab + b^2$.

Let's identify $a$ and $b$ in our expression:

• Here, $a = x$ and $b = x^2$.

Applying the formula:

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

Calculating each part, we get:

• $(x)^2 = x^2$
• $-2(x)(x^2) = -2x^3$
• $(x^2)^2 = x^4$

Combining these results, the expression simplifies to:

$x^4 - 2x^3 + x^2$

Therefore, the expanded form of $(x-x^2)^2$ is $\boxed{x^4 - 2x^3 + x^2}$.