Power of a Product: Applying the formula

To solve the expression $\left(2\times11\right)^5$, 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 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: $10^4 \times 3^4$
  This is exactly what we found using the power of a product rule

Thus, the solution is that all answers are correct.

To solve the problem, we need to apply the exponent rule known as the Power of a Product, 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:

$9^4 \times 7^4$

This demonstrates the application of the Power of a Product rule successfully, showing detailed steps and the correct application of exponential laws.

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 the given 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 choices given:

• 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 "a' + b' are correct," which indicates the validity of both $6^4 \times 8^4$ and $48^4$ as representations of $(6 \times 8)^4$.

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 into the rule:
  $(5 \times 7)^3 = 5^3 \times 7^3$

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

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.

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 is given by $(a \times b)^n = a^n \times b^n$.

• 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 "B+C are correct". This acknowledges that "B" expresses $15^3$, and "C" independently describes the valid expanded expression, both right solutions for different cases of interpretations.

To solve the problem $\left(4\times2\right)^2$, we need to apply the rule of exponents known as the "Power of a Product". 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 $4^2 \times 2^2$.

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: $2^2 \times 3^2$.

To solve the question, we apply the rule of exponents known as the Power of a Product. The formula states that for any real numbers $a$ and $b$, and an integer $n$:

• $(a \times b)^n = a^n \times b^n$

Given the expression $(2 \times 4)^{10}$, we can identify:

• $a = 2$
• $b = 4$
• $n = 10$

Now, applying the formula:

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

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

To solve the expression $\left(5\times6\right)^9$, we 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 $5^9\times6^9$.

The given expression to solve is $\left(13\times4\right)^6$. This expression involves raising a product to a power. According to the exponent rule, which is the Power of a Product Rule, the property can be stated as follows:

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

Mathematically, for any numbers $a$ and $b$, and a positive integer $n$, it is written as: $(a \times b)^n = a^n \times b^n$.

Applying this rule to our expression:

$\left(13 \times 4\right)^6$ becomes $13^6 \times 4^6$.

Thus, the expression $\left(13 \times 4\right)^6$ simplifies to $13^6 \times 4^6$ following the Power of a Product Rule.

The given expression is $(12 \times 3)^5$. According to the Power of a Product rule, which states that $(a \times b)^n = a^n \times b^n$, we 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 $12^5 \times 3^5$.

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.).

To solve the problem$(2\times7\times5)^3$, we need to apply the Power of a Product rule of exponents. 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 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 $2^3 \times 7^3 \times 5^3$.

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.

To solve the given expression, we need to apply the rule known as the Power of a Product. This rule states that the product of two numbers raised to the same power can be expressed as the product of those numbers raised to that power. Mathematically, it's represented as: $a^n \times b^n = (a \times b)^n$.

In our given problem, we have $11^6 \times 4^6$.

• Here, the base numbers are 11 and 4, and both are raised to the 6th power.
• According to the Power of a Product rule, we can combine these into a single expression: $(11 \times 4)^6$.

Thus, the expression $11^6 \times 4^6$ can be rewritten as $(11 \times 4)^6$. This is the simplified or equivalent expression.

To solve the expression $8^7 \times 10^7$, we can use the power of a product rule, which states that $a^m \times b^m = (a \times b)^m$. Here,$a = 8$ and $b = 10$, and both are raised to the same power $m = 7$.

Following these steps:

• Identify the base numbers and the common exponent: Here, the base numbers are $8$ and $10$, and the common exponent is $7$.

• Apply the power of a product rule: Instead of multiplying $8^7$ and $10^7$ directly, we apply the rule to get $(8 \times 10)^7$.

• This simplifies to $(80)^7$.

Thus, the rewritten expression is $\left(8 \times 10\right)^7$.

We are given the expression: $2^3 \times 4^3$ and need to express it as a single term using the power of a product rule.

The power of a product rule states that for any non-zero numbers $a$ and $b$, and an integer $n$, $(a \times b)^n = a^n \times b^n$.

To apply the inverse formula, which is converting two separate powers into a product raised to a power, we look for terms that can be combined under a single exponent. Observe that:

• Both terms $2^3$ and $4^3$ have the same exponent.

• This allows us to combine them into a single expression: $(2 \times 4)^3$.

Therefore, according to the power of a product rule applied inversely, the expression $2^3 \times 4^3$ can be rewritten as $(2 \times 4)^3$.

To solve the given expression, we need to apply the 'Power of a Product' rule in exponentiation. This rule states that for any numbers $a$ and $b$:

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

In this problem, the base numbers are 3 and 4, and the exponent is 4. Therefore, we can rewrite the expression $3^4 \times 4^4$ using the power of a product rule:

• Identify the bases: 3 and 4.

• Identify the common exponent: 4.

• Apply the rule: $(3 \times 4)^4$

Thus, the expression $3^4 \times 4^4$ can be rewritten as $(3 \times 4)^4$.