Power of a Product: Applying the formula

Step 1: We start with the expression $\left(20 \times 5\right)^7$.
Step 2: We'll apply the power of a product rule, which states $(a \times b)^n = a^n \times b^n$. This gives us: $\left(20 \times 5\right)^7 = 20^7 \times 5^7$ .
Step 3: To verify, notice that both $20^7 \times 5^7$ and $5^7 \times 20^7$ involve the same expression due to the commutative property of multiplication. Also, we can rewrite $\left(20 \times 5\right)$ as $100$, leading to another form: $\left(20 \times 5\right)^7 = 100^7$
Thus, both $20^7 \times 5^7$, $5^7 \times 20^7$, and $100^7$ are equivalent expressions for $\left(20 \times 5\right)^7$.

Therefore, the correct answer choice is (d) "All answers are correct."

The expression in question is $\left(13\times4\right)^6$. This expression involves raising a product to a power and requires the application of the power of a product exponent rule, which states:

• When you have a product raised to an exponent, you can distribute the exponent to each factor in the product separately.

Mathematically, any numbers $a$ and $b$ and a positive integer $n$ can be written as$(a \times b)^n = a^n \times b^n$.

Applying this rule to our expression, $\left(13 \times 4\right)^6$ becomes $13^6 \times 4^6$.

Therefore, the expression $\left(13 \times 4\right)^6$ simplifies to $13^6 \times 4^6$.

The given expression is $(12 \times 3)^5$. The power of a product rule states that $(a \times b)^n = a^n \times b^n$. We will apply this formula to the expression.

• Firstly, identify the base of the power as the product $12 \times 3$.

• Secondly, recognize that the exponent applied to this product is 5.

• According to the rule, the power of a product can be distributed to each factor in the product, which means: $(12 \times 3)^5 = 12^5 \times 3^5$.

Therefore, the expression $(12 \times 3)^5$ corresponds to $12^5 \times 3^5$.

The given expression is $\left(2\times3\right)^2$. We need to apply the rule of exponents known as the "Power of a Product." This rule states that when you have a product raised to an exponent, you can apply the exponent to each factor in the product individually. Mathematically, this is expressed as: $(a \times b)^n = a^n \times b^n$.

In this case, the expression $\left(2\times3\right)^2$ follows this rule with $a = 2$ and $b = 3$, and $n = 2$.

• First, apply the exponent to the first factor: $2^2$.
• Next, apply the exponent to the second factor: $3^2$.

Therefore, by applying the "Power of a Product" rule, the expression becomes: $2^2 \times 3^2$.

The problem requires us to simplify the expression $(5 \times 7)^3$ using the power of a product rule.

The power of a product rule states that for any numbers $a$ and $b$, and any integer $n$, the expression $(a \times b)^n$ can be expanded to $a^n \times b^n$.

Applying this rule to the given expression:

• Identify the values of $a$ and $b$ as $5$ and $7$, respectively.

• Identify $n$ as $3$.

• Substitute using the rule:
  $(5 \times 7)^3 = 5^3 \times 7^3$

The simplified expression is therefore $5^3 \times 7^3$.

To solve the expression $\left(5\times6\right)^9$, we need to apply the rule for the power of a product. This rule states that when you raise a product to a power, it is equivalent to raising each factor in the product to the same power. Mathematically, this is expressed as:

• $(a \times b)^n = a^n \times b^n$

In this specific problem, the factors are $5$ and $6$ and the exponent is $9$. Applying the power of a product rule, we get:

• $\left(5\times6\right)^9 = 5^9 \times 6^9$

Therefore, the correct expression that corresponds to $\left(5\times6\right)^9$ is $5^9\times6^9$.

To solve the expression, we can apply the rule for the power of a product, which states that$\left(a \times b\right)^n = a^n \times b^n$.

In this case, our expression is $\left(2\times11\right)^5$, where$a = 2$ and $b = 11$, and $n = 5$.

Applying the power of a product rule gives us:

• $a^n = 2^5$

• $b^n = 11^5$

Therefore, $\left(2\times11\right)^5 = 2^5 \times 11^5$.

To solve the problem of rewriting the expression $(6 \times 8)^4$, we will apply the power of a product rule. This rule states that $(a \times b)^n = a^n \times b^n$.

Let's apply this rule to our expression:

$(6 \times 8)^4$ can be rewritten as $6^4 \times 8^4$ by applying the Power of a Product rule.

Now, let's consider the available options:

• Choice 1: $6^4 \times 8^4$ - This choice directly corresponds to our rewritten expression using the power of a product rule.

• Choice 2: $48^4$ - This represents the product calculated first ($6 \times 8 = 48$) and then raised to the fourth power. This is also a valid representation since $(6 \times 8)^4$ is equivalent to $48^4$.

• Choice 3: $6^4 \times 8$ - This is incorrect because it does not correctly apply the power of a product rule.

Therefore, the correct answer according to the given choices is that (a) and (b) are correct.

To solve the problem $\left(4\times2\right)^2$, we need to apply the power of a product exponent rule. This rule states that $(ab)^n = a^n \times b^n$.

Here, $a = 4$, $b = 2$ and $n = 2$.

• Step 1: Apply the power of a product rule: $\left(4 \times 2\right)^2 = 4^2 \times 2^2$.

Thus, the expression $\left(4\times2\right)^2$ is equivalent to $4^2 \times 2^2$.

To solve the problem, we need to apply the power of a product exponent rule. This rule states that when you raise a product to a power, it's the same as raising each factor to that power. In mathematical terms, if you have $(abc)^n$, it is equivalent to $a^n \times b^n \times c^n$.

Let's apply this rule step by step:

Our original expression is: $(2 \times 7 \times 5)^3$.

We first identify the factors inside the parentheses as $2$, $7$, and $5$.

According to the Power of a Product rule, we can distribute the exponent$3$ to each factor:

First, raise $2$ to the power of $3$ to get $2^3$.

Then, raise $7$ to the power of $3$ to get $7^3$.

Finally, raise $5$ to the power of $3$ to get $5^3$.

Therefore, the expression $(2 \times 7 \times 5)^3$ simplifies to $2^3 \times 7^3 \times 5^3$.

To solve the problem, we need to apply the power of a product rule for exponents, which states that if you have a product raised to an exponent, you can apply the exponent to each factor in the product individually.

The general form of this rule is:

$(a \times b)^n = a^n \times b^n$

According to this formula, when we have the expression$(9 \times 7)^4$ we apply the exponent 4 to each factor within the parentheses.

This process results in:

• Raising 9 to the power of 4: $9^4$

• Raising 7 to the power of 4: $7^4$

Therefore, the expression simplifies to:

$9^4 \times 7^4$

To solve the question, we need to apply the power of a product exponent rule. The formula states that for any real numbers $a$ and $b$, and any integer$n$:

• $(a \times b)^n = a^n \times b^n$

Looking at our expression, we can see that:

• $a = 2$

• $b = 4$

• $n = 10$

Now, if we apply the formula:

• $(2 \times 4)^{10} = 2^{10} \times 4^{10}$

Therefore, the expression $(2 \times 4)^{10}$ is equivalent to $2^{10} \times 4^{10}$.

To solve this problem, we'll apply the power of a product rule to the expression $(10 \times 3)^4$.

• Step 1: Identify the expression.
  The given expression is $(10 \times 3)^4$.

• Step 2: Apply the power of a product law.
  According to the rule, $(a \times b)^n = a^n \times b^n$, our expression becomes:
  $(10 \times 3)^4 = 10^4 \times 3^4$.

• Step 3: Evaluate the choices:
  - First choice: $3^4 \times 10^4$
  Rearranging terms, this is equivalent to $10^4 \times 3^4$. Therefore, it matches our transformed expression.
  - Second choice: $30^4$
  Since $30 = 10 \times 3$, $(10 \times 3)^4 = 30^4$. This simplifies to the same expression.
  - Third choice: $10^4 \times 3^4$

  Therefore, the solution is that all answers are correct.

To solve this problem, we'll clarify our understanding and execution as follows:

• Identify the given expression: $(5 \times 3)^3$.

• Apply the relevant formula: the power of a product rule —$(a \times b)^n = a^n \times b^n$.

• Execution: Rewrite $(5 \times 3)^3$ as $5^3 \times 3^3$.

Let's walk through the solution:

Initially, we have the expression $(5 \times 3)^3$. We recognize that by the power of a product rule, this can be rewritten as $5^3 \times 3^3$.

Next, let's verify the choices:
- $5^3 \times 3$ is incorrect as the exponent doesn't apply to both factors.
- $15^3$ represents the base simplified ($5\times3=15$).
- $5^3 \times 3^3$ matches exactly what we derived earlier, making this the correct expression resulting from the power rule.

Therefore, the correct choice is option is "Answers (b) and (c) are correct". This acknowledges that "B" expresses $15^3$, and "C" independently describes the valid expanded expression, both solutions for different cases of interpretations.

To solve this problem, we will apply the "power of a product" rule, which states that if you have a product raised to a power, each factor of the product is raised to that power. We have:

$\left(25 \times 4\right)^3 = 25^3 \times 4^3$

Using the commutative property of multiplication, this can alternatively be written as:

$4^3 \times 25^3$

Additionally, observing that $25 \times 4 = 100$, we also have:

$(100)^3$

Thus, all the given choices are equivalent expressions for the given problem based on the power of a product and basic arithmetic simplification.

• $25^3 \times 4^3$

• $4^3 \times 25^3$

• $\left(100\right)^3$

Therefore, the correct answer is that all answers are correct.

We are given the expression $\left(2\times6\right)^3$ and need to simplify it using the power of a product rule in exponents.

The power of a product rule states that when you have a product inside a power, you can apply the exponent to each factor in the product individually. In mathematical terms, the rule is expressed as:

• $(a \cdot b)^n = a^n \cdot b^n$

Applying this to our expression, we have:

$\left(2\times6\right)^3 = 2^3\times6^3$

This means that each term inside the parentheses is raised to the power of 3 separately.

Therefore, the expression $\left(2\times6\right)^3$ simplifies to $2^3\times6^3$ as per the power of a product rule.

We begin by using the power rule for parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

That is, the power applied to a product inside parentheses, is applied to each of the terms within, when the parentheses are opened.

We then apply the above rule to the problem:

$(2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2$

Therefore, the correct answer is option d.

Note:

From the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms inside parentheses (as formulated above), then it is also valid for a power over several terms of the product inside parentheses (for example - three terms, etc.).

We use the law of exponents for a power that is applied to parentheses in which terms are multiplied:

$(x\cdot y)^n=x^n\cdot y^n$

We apply the rule to the problem:

$(9\cdot2\cdot5)^3=9^3\cdot2^3\cdot5^3$

When we apply the power within parentheses to the product of the terms we do so separately and maintain the multiplication,

Therefore, the correct answer is option B.

Let us begin by using the law of exponents for a power that is applied to parentheses in which terms are multiplied:

$(x\cdot y)^n=x^n\cdot y^n$

We apply the rule to our problem:

$(x\cdot4\cdot3)^3= x^3\cdot4^3\cdot3^3$

When we apply the power to the product of the terms within parentheses, we apply the power to each term of the product separately and keep the product,

Therefore, the correct answer is option C.

The expression in question is $(11 \times 15 \times 4)^6$.

Using the power of a product rule, we know that any numbers $a$, $b$, and $c$ can be written as$(a \times b \times c)^n = a^n \times b^n \times c^n$.

Applying this, we get:

$(11 \times 15 \times 4)^6 = 11^6 \times 15^6 \times 4^6$

Verify the multiple-choice options:
- Option 1: Clearly represents the expression as $11^6 \times 15^6 \times 4^6$, so this is correct.
- Option 2: If we combine $15 \times 4 = 60$, the expression becomes $(11 \times 60)^6$, which matches $11^6 \times 60^6$, therefore correct.
- Option 3: If we combine $11 \times 4 = 44$, the expression becomes $(44 \times 15)^6$, which aligns with $44^6 \times 15^6$, therefore correct.

Since all three expressions are validated as equivalent to the original expression when simplified appropriately, all answers are correct.