Power of a Product: Number of terms

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.

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

We need to apply the power of a product rule to our expression.

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:

• $8^7$

• $5^7$

• $2^7$

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

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

The problem requires simplifying the expression $(2\times5\times4)^7$ using the power of a product exponent rule. According to the 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 $\left(12 \times 5 \times 4\right)^{10}$, we apply the power of a product rule of exponents, which 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 $\left(8\times7\times3\right)^8$, 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.

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

To solve the expression, 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:

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

To solve the given problem, we'll utilize the power of a product rule. This rule states that:

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

Let's apply this rule to the expression $(9 \times 6 \times 8)^5$:

$(9 \times 6 \times 8)^5 = 9^5 \times 6^5 \times 8^5$

It's important to note that when using this rule, the order of terms under multiplication does not affect the result due to the commutative property of multiplication. Thus, any permutation of the factors yields the same mathematical value. Hence, the expression can also be written as:

• $8^5 \times 6^5 \times 9^5$
• $8^5 \times 9^5 \times 6^5$
• $9^5 \times 8^5 \times 6^5$

Upon reviewing the answer choices, each choice represents a valid permutation of $9^5 \times 6^5 \times 8^5$.

This leads us to conclude that all given options are correct, including the explicit choice stating this fact.

Therefore, the solution to the problem is: All answers are correct.

To solve the problem $\left(11 \times 6 \times 5\right)^6$, we'll apply the power of a product exponent rule. This rule states that when you raise a product to an exponent, you can distribute the exponent to each factor within the product.

Step 1: The given expression is $\left(11 \times 6 \times 5\right)^6$.

Step 2: According to the power of a product rule, $(a \times b \times c)^n = a^n \times b^n \times c^n$.

Applying this to our expression we get:

$\left(11 \times 6 \times 5\right)^6 = 11^6 \times 6^6 \times 5^6$

Step 3: Since multiplication is commutative (i.e., the order of multiplication doesn't affect the result), we can rearrange the terms as follows:

$11^6 \times 6^6 \times 5^6 = 5^6 \times 6^6 \times 11^6$

Both expressions $11^6 \times 6^6 \times 5^6$ and $5^6 \times 6^6 \times 11^6$ are equivalent and correctly represent the expanded form of the original expression.

Therefore, the correct corresponding expressions are both $11^6 \times 6^6 \times 5^6$ and $5^6 \times 6^6 \times 11^6$.

Conclusion: Answers (a) and (b) are correct.

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 $c$ with 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 $4^9 \times 10^9 \times 7^9$.

To solve the problem, we need to apply the power of a product rule of exponents, which 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 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 this problem, let's expand the expression $(2 \cdot 6 \cdot 3)^9$.

Step 1: Apply the power of a product rule: $(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$.

Step 2: Identify the factors. In this case, we have three factors: $2$, $6$, and $3$.

Step 3: Apply the exponent 9 to each factor:

• $2^9$

• $6^9$

• $3^9$

Step 4: Combine these into the fully expanded form:

$2^9 \cdot 6^9 \cdot 3^9$.

Review the answer options and note that there is no choice that matches this correct form.

Therefore, none of the answer choices are correct.

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

• Step 1: Apply the power of a product rule to the given expression.

• Step 2: Evaluate each choice to see if it matches the original expression.

Let's work through the problem:
Step 1: Start with the expression $(4 \times 10 \times 7)^9$. According to the power of a product rule, we can express this as $4^9 \times 10^9 \times 7^9$.

Step 2: Evaluate each answer choice:

• Choice 1: $(40 \times 7)^9$ -> Simplifies to $40^9 \times 7^9$, does not match directly since $10^9$ is missing.

• Choice 2: $(4 \times 70)^9$ -> Simplifies to $4^9 \times 70^9$, does not match directly since $7^9$ is extra.

• Choice 3: Answers (a) + (b) are correct -> Considering the intent behind (a) and (b), mixes the choices recognised as potentially correct via simplifications.

• Choice 4: $4 \times 10^9 \times 7^9$ -> is incorrect as it has $4$ not raised to the power 9.

Therefore, based on calculation errors observed in each option, the correct conclusion for complex-simplified choices is: Answers (a) and (b) are correct.

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 need to apply the power of products exponent rule. 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:

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

We can apply the power of a product rule to our expression:

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

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.

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.

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

In this problem, we recognize that 8, 10, and 16 all have the same exponent of 7. Therefore 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 $8^7\times10^7\times16^7$.

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 then 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:

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