Power of a Product: Applying the formula

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 $12^5 \times 3^5$.

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 $\left(2\times3\right)^2$. 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 $2^{10} \times 4^{10}$.

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 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 $4^2 \times 2^2$.

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 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: $5^3 \times 7^3$.

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 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 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 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.

To solve the question, we need to apply the power of a product rule from exponents. This rule states that when a product is raised to an exponent, we can apply the exponent to each factor within the product individually. Mathematically, the rule is expressed as:

$(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$.

Now, we identify the components in the given expression:

• The expression inside the parentheses is $2 \times 6 \times 8$.
• The exponent applied to this product is $4$.

Applying the exponent to each factor gives us:

• Apply the exponent 4 to the factor 2: $2^4$.
• Apply the exponent 4 to the factor 6: $6^4$.
• Apply the exponent 4 to the factor 8: $8^4$.

Therefore, the expression $(2 \times 6 \times 8)^4$ is transformed into:

$2^4 \times 6^4 \times 8^4$.

This matches the correct answer provided: $2^4 \times 6^4 \times 8^4$.

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 $\left(8 \times 10\right)^7$.

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.

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 $(2 \times 4)^3$.

The problem given is to simplify the expression $8^5 \times 9^5$. This problem can be solved by applying the exponent rules, specifically the Power of a Product rule.

Step-by-step solution:

• According to the Power of a Product rule, when two expressions with the same exponent are multiplied, the product can be written as a single power expression:
  $a^n \times b^n = (a \times b)^n$.
• Using this rule, we can rewrite the given expression $8^5 \times 9^5$ as a single power:
  $(8 \times 9)^5$.
• Therefore, the expression simplifies to:
  $(72)^5$.
• We have represented $8^5 \times 9^5$ as $(72)^5$, which is its corresponding expression according to the Power of a Product rule.

By recognizing that $8^5 \times 9^5$ can be expressed as a single power of $72$, we confirm that the problem demonstrates the application of the Power of a Product rule. All provided conversions of this expression are mathematically correct, warranting the conclusion "All answers are correct".

To solve the expression $\left(12 \times 5 \times 4\right)^{10}$, we apply the rule of exponents known as the "Power of a Product". This rule states that when you have a product inside a power, you can apply the exponent to each factor in the product separately. This can be expressed by the formula:

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

In the given expression, the base is the product $12 \times 5 \times 4$ and the exponent is $10$.

Therefore, according to the power of a product rule, the expression can be rewritten by raising each individual base to the power of $10$:

• Raise 12 to the 10th power: $12^{10}$

• Raise 5 to the 10th power: $5^{10}$

• Raise 4 to the 10th power: $4^{10}$

Thus, the expression $\left(12 \times 5 \times 4\right)^{10}$ simplifies to:

$12^{10} \times 5^{10} \times 4^{10}$

This shows the application of the Power of a Product rule for exponents by distributing the 10th power to each term within the parentheses.

To solve the expression $\left(16\times2\times3\right)^{11}$, we will use the Power of a Product rule. According to this rule, when you have a product raised to an exponent, you can distribute the exponent to each factor in the product. Mathematically, this is expressed as:

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

• In our expression, $a = 16$, $b = 2$, and $c = 3$.

Applying the Power of a Product formula to our expression gives:

$(16 \times 2 \times 3)^{11} = 16^{11} \times 2^{11} \times 3^{11}$

This shows that each factor inside the parentheses is raised to the power of 11, which is consistent with the provided answer:

$16^{11}\times2^{11}\times3^{11}$