Power of a Product: Number of terms

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 $\left(12 \times 5 \times 4\right)^{10}$, 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:

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

We need to apply the rule of exponents known as the Power of a Product. According to this rule, when we raise a product to a power, we can raise each individual term in the product to that power.

Mathematically, this is expressed as:

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

Given the expression:

$\left(2 \times 5 \times 6\right)^{15}$

We can apply the Power of a Product rule:

• Raise each factor inside the parentheses to the power of 15.

This gives us:

$2^{15} \times 5^{15} \times 6^{15}$

Therefore, the expression is $2^{15} \times 5^{15} \times 6^{15}$.

To solve the problem, we need to apply the rule of exponents known as the "power of a product." This rule states that when you raise a product to a power, you can distribute the exponent to each factor in the product.

Let's break it down with the given problem:

We have the expression $\left(3\times7\times9\right)^8$. According to the power of a product rule, this expression can be rewritten by raising each individual factor inside the parentheses to the power of 8:

• Take the number 3 and raise it to the power of 8: $3^8$
• Take the number 7 and raise it to the power of 8: $7^8$
• Take the number 9 and raise it to the power of 8: $9^8$

Now, we can use the rule to rewrite the original expression as the product of these terms:

$3^8\times7^8\times9^8$

This is the expression you obtain when you apply the power of a product rule to $\left(3\times7\times9\right)^8$.

To solve the problem, we need to apply the power of a product rule of exponents. This rule states that $(a \times b \times c)^n = a^n \times b^n \times c^n$ for any numbers $a$, $b$, and $c$, and an exponent $n$.

Let's apply this rule to the given expression: $\left(4\times10\times7\right)^9$:

• Identify each factor inside the parentheses: 4, 10, and 7.

• The exponent applied is 9.

• Apply the rule: Each factor inside the parentheses is raised to the 9th power.

This gives us the expression: $4^9 \times 10^9 \times 7^9$.

Therefore, the final expression is: $4^9 \times 10^9 \times 7^9$

The problem involves applying the Power of a Product rule in exponents. This rule states that when you raise a product to an exponent, you can apply the exponent to each factor in the product separately. Mathematically, this rule is expressed as: $(a \times b \times c)^n = a^n \times b^n \times c^n$.

Given the expression: $(8 \times 5 \times 2)^7$, we need to apply the Power of a Product rule:

First, identify each individual factor in the product:

• Factor 1: $8$
• Factor 2: $5$
• Factor 3: $2$

Now, apply the exponent $7$ to each factor:

• $8^7$
• $5^7$
• $2^7$

So the expression $(8 \times 5 \times 2)^7$ simplifies to:

$8^7 \times 5^7 \times 2^7$

To solve the expression $\left(8\times7\times3\right)^8$, we can use the "power of a product" rule. This rule states that when raising a product to an exponent, you can apply the exponent to each factor within the parentheses.

So, according to the rule:

$\left(8\times7\times3\right)^8 = 8^8 \times 7^8 \times 3^8$

Each of the factors: 8, 7, and 3, is independently raised to the power of 8.

This approach allows us to separate the original power into the power of each individual factor, making the expression equivalent to multiplying each of these results together.

Therefore, the corresponding expression that equals $\left(8\times7\times3\right)^8$ is:

• $8^8 \times 7^8 \times 3^8$

• Each factor separately raised to the 8th power, then multiplied together.

Ultimately, all answers similar to this transformation are correct, as they apply the correct exponent rules.

We begin by noting that the given expression is $\left(9\times10\times7\right)^5$. Our task is to expand this expression using the power of a product rule.

The power of a product rule states that for any real numbers $a$, $b$, and $c$ and a positive integer $n$, $(a \times b \times c)^n = a^n \times b^n \times c^n$.

Applying this rule to the given expression, we set $a = 9$, $b = 10$, and $c = 7$, and $n = 5$.

By substituting these into the power of a product formula, we have:

• $a^n = 9^5$

• $b^n = 10^5$

• $c^n = 7^5$

Therefore, the expression $\left(9\times10\times7\right)^5$ expands to:

$9^5\times10^5\times7^5$

We use the power law for multiplication within parentheses:

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

We apply it to the problem:

$(3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4$

Therefore, the correct answer is option b.

Note:

From the formula of the power property mentioned above, we understand that it refers not only to two terms of the multiplication within parentheses, but also for multiple terms within parentheses.

We use the power law for multiplication within parentheses:

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

We apply it to the problem:

$(4\cdot7\cdot3)^2=4^2\cdot7^2\cdot3^2$

Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we understand that we can apply it not only to the multiplication of two terms within parentheses, but is also for multiple terms within parentheses.

The problem requires simplifying the expression $(2\times5\times4)^7$ using the power of a product rule. According to the exponent rules, specifically the power of a product rule, we know that:

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

This means when we have a product raised to an exponent, each factor in the product is raised to that exponent. So let's apply this rule to the given expression:

• First, identify the terms inside the parentheses: 2, 5, and 4.

• Next, apply the exponent 7 to each term:

  • $2^7$ – The first term 2 is raised to the power of 7.

  • $5^7$ – The second term 5 is raised to the power of 7.

  • $4^7$ – The third term 4 is raised to the power of 7.

Therefore, the expression $(2\times5\times4)^7$ simplifies to:

$2^7\times5^7\times4^7$

To solve the expression $20^{10} \times 4^{10} \times 2^{10}$, we can apply the Power of a Product rule for exponents. According to this rule, for any numbers $a$, $b$, and $c$ raised to the power of $n$, we have:

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

In this case, we have:

• $a = 20$

• $b = 4$

• $c = 2$,

• $n = 10$.

Thus, we can write:

$20^{10} \times 4^{10} \times 2^{10} = (20 \times 4 \times 2)^{10}$

The given expression is $2^9 \times 4^9 \times 7^9$.

We need to simplify this expression by using the exponent rule for the power of a product. The rule states that $(a \times b \times c)^n = a^n \times b^n \times c^n$, which can be inverted to simplify a product of terms with the same exponent.

Thus, the expression $2^9 \times 4^9 \times 7^9$ can be rewritten as $(2 \times 4 \times 7)^9$.

Breaking it down:

• Identify the common exponent, which is $9$.
• Combine the bases under a single power:
• The bases are $2, 4,$ and $7$.
• Apply the exponent rule: $2^9 \times 4^9 \times 7^9 = (2 \times 4 \times 7)^9$.

Therefore, the corresponding expression is $\left(2\times4\times7\right)^9$.

The given expression is $8^7\times10^7\times16^7$. We need to apply the power of a product rule for exponents. This rule states that for any numbers $a$, $b$, and $c$, if they have the same exponent $n$, then $a^n \times b^n \times c^n = (a\times b \times c)^n$.

In this problem, we recognize that 8, 10, and 16 all have the same exponent of 7, thus, we can apply the rule directly:

• $8^7 \times 10^7 \times 16^7$
• Applying the power of a product rule:
• $(8 \times 10 \times 16)^7$

This simplified form matches the pattern we recognize from the power of a product rule, verifying that $(8\times10\times16)^7$ is indeed the correct transformation of the original expression $8^7\times10^7\times16^7$, thereby confirming our answer is correct.

To solve the expression $9^9 \times 8^9 \times 2^9$, we will apply the "Power of a Product" rule in the exponent rules. This rule states that if you have a product of terms all raised to the same exponent, it can be rewritten as the product itself raised to that exponent. The formula is given by:

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

In our problem, we can identify the terms as:

• $a = 9$

• $b = 8$

• $c = 2$

• $n = 9$

Applying the formula, we can convert the product of powers into a single power:

$9^9 \times 8^9 \times 2^9 = (9 \times 8 \times 2)^9$

The goal is to apply the power of a product rule by transforming the expression $5^8 \times 8^8 \times 10^8$ into the form $(a \times b \times c)^n$.

Let's begin by identifying the terms involved:

• The expression consists of three separate terms, each raised to the 8th power: $5^8$, $8^8$, and $10^8$.

According to the power of a product rule, $(a \times b \times c)^n = a^n \times b^n \times c^n$. Therefore, given that the exponents are the same, $n = 8$, we can reverse this process.

• The original expression is $5^8 \times 8^8 \times 10^8$.

We can consolidate this into a single term by combining the bases under the same exponent:

Thus, $5^8 \times 8^8 \times 10^8 = (5 \times 8 \times 10)^8$.

Therefore, the corresponding expression is:

$\left(5\times8\times10\right)^8$

To solve the expression $5\times3^5\times11^5$, we can apply the rule of exponents known as the Power of a Product rule. This rule states that for any integers $a$, $b$, and $n$, $(a\times b)^n = a^n \times b^n$.

Step 1: Analyze the expression
The expression we have is $5\times3^5\times11^5$.

Step 2: Apply the Power of a Product rule
Notice that both 3 and 11 are raised to the power of 5. We can use the inverse of the Power of a Product formula to combine these terms:

• $3^5 \times 11^5$ can be written as$(3 \times 11)^5$

Step 3: Rewrite the expression
Therefore, the expression $5\times3^5\times11^5$ becomes $5\times(3\times11)^5$.

By applying the Power of a Product rule, we have determined that the equivalent expression for the given problem is $5\times(3\times11)^5$.

To solve the given expression, we need to apply the power of a product rule from exponent rules. This rule states that when you have a product of numbers raised to the same power, you can combine them under one exponent. The formula is as follows:

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

In our problem, we are given:
$7^{11}\times3^{11}\times8$

We notice that $7^{11}$ and $3^{11}$ are raised to the same power, 11. Therefore, according to the power of a product rule, we can combine them:

• Identify the terms with the same power: $7^{11}$ and $3^{11}$.

• Combine the terms under a single exponent: $(7 \times 3)^{11}$.

This means:

$7^{11}\times3^{11} = (7\times3)^{11}$

Thus, our simplified expression now looks like this:

$(7\times3)^{11}\times8$

We use the power property for an exponent that is applied to a set parentheses in which the terms are multiplied:

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

We apply the law in the problem:

$(7\cdot4\cdot6\cdot3)^4=7^4\cdot4^4\cdot6^4\cdot3^4$

When we apply the exponent to a parentheses with multiplication, we apply the exponent to each term of the multiplication separately, and we keep the multiplication between them.

Therefore, the correct answer is option a.