Power of a Product: Inverse formula

The problem given is to simplify the expression $8^5 \times 9^5$. This problem can be solved by applying the power of a product rule exponent 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 of the expressions are mathematically the same and therefore the correct answer is (d) "All answers are correct".

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 $(80)^7$.

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

To solve the expression, we need to apply the power of a product exponent rule. 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 $(11 \times 4)^6$.

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 like so:

• 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 $(3 \times 4)^4$.

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 any non-zero numbers $a$ and $b$ and an integer $n$ can be written as $(a \times b)^n = a^n \times b^n$.

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. Note:

• 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 $(2 \times 4)^3$.

We are given the expression $5^5 \times 6^5$.
According to the power of a product rule, $(a \times b)^n = a^n \times b^n$. Therefore, we can rewrite the expression as $(5 \times 6)^5$ or $(6 \times 5)^5$.

Both expressions $(5 \times 6)^5$ and $(6 \times 5)^5$ are mathematically equivalent. Therefore, the solution to the problem is that answers (a) and (b) are correct.

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:

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

To address this problem, let's use the power of a product rule:

We start with $11^6 \times 10^6 \times 12^6$.

By the power of a product rule, we can combine these into a single expression: $(11 \times 10 \times 12)^6$.

This equation satisfies the choices given, as all representations like $(11 \times 10 \times 12)^6$, $(10 \times 11 \times 12)^6$, and $(12 \times 10 \times 11)^6$ are equivalent due to the commutative property of multiplication.

Thus, all answers provided are correct.

To answer the question, we need to apply the power of a product exponent rule. 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:

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

To simplify the expression $2^7 \times 3^7 \times 10^7$, we can follow these steps:

• Step 1: Apply the power of a product rule to combine $2^7 \times 3^7$ into $(2 \times 3)^7$.

• Step 2: Recognize that $2 \times 3 = 6$, so the expression becomes $6^7 \times 10^7$.

• Step 3: Again, apply the power of a product rule to combine $6^7 \times 10^7$ into $(6 \times 10)^7$.

• Step 4: Calculate $6 \times 10 = 60$, resulting in $60^7$.

Therefore, the simplified form of the expression is $60^7$. Additionally, since all the provided answer choices ($60^7$, $6^7 \times 10^7$, and $2^7 \times (3 \times 10)^7$) represent equivalent forms of the original expression, all answers are correct.

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 we can state:

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

In this case, we have:

• $a = 20$

• $b = 4$

• $c = 2$

• $n = 10$

Thus, we can write:

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

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: Analyse 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 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:

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

To solve the problem $9^{11}\times7^{11}\times6^{11}=\text{ }$, we will apply the power of a product rule.

Step 1: Identify the expression and common exponent
The expression given is $9^{11} \times 7^{11} \times 6^{11}$. Notice that all three terms share the common exponent 11.

Step 2: Apply the Power of a Product rule
According to the power of a product rule, where you have multiple terms each raised to the same power, you can rewrite the expression as a single product raised to that common power. This means:

$(9 \times 7 \times 6)^{11}$

This expression consolidates the original terms under a single exponent.

Step 3: Verify the form of the solution
The choices provided show the expression in a generalized form without calculating the product. Hence, the expression can be represented as:

$(9 \times 7 \times 6)^{11}$ or $(6 \times 7 \times 9)^{11}$ or $(7 \times 6 \times 9)^{11}$

Conclusion:
Therefore, any of the options where the bases are multiplied together under the common exponent 11 correctly represent the simplified expression. Thus, the answer is "All of the above".

To address this mathematical problem, we need to rewrite the expression $\frac{1}{3^4 \times 12^4}$ using exponent properties. Specifically, we'll use the property that tells us $\frac{1}{a^n} = a^{-n}$.

Let's perform the necessary steps:

• Step 1: Apply the formula to each component in the denominator. For $\frac{1}{3^4}$, we have $3^{-4}$ using the rule $\frac{1}{a^n} = a^{-n}$.
• Step 2: Similarly, for $\frac{1}{12^4}$, we apply the rule to get $12^{-4}$.
• Step 3: Combine these results using multiplication: $3^{-4} \times 12^{-4}$.

By performing these transformations, we can confirm that the expression $\frac{1}{3^4 \times 12^4}$ is equivalent to $3^{-4} \times 12^{-4}$.

Therefore, the correct expression is $3^{-4} \times 12^{-4}$, which is choice 3 in the multiple-choice options provided.

To reduce the expression $a^8 \times b^8 \times c^8$, we can apply the Power of a Product Rule, which states that when multiplying powers with the same exponent across different bases, we can combine them into a single power. Specifically, this rule is written as:

$(x^m \times y^m \times z^m) = (x \times y \times z)^m.$

Applying this rule to our expression $a^8 \times b^8 \times c^8$, we identify $x = a$, $y = b$, and $z = c$, all with the exponent $m = 8$. Therefore, we can simplify the expression to:

$(a \times b \times c)^8.$

Thus, the reduced form of the given expression is:

$(a \times b \times c)^8$.

To solve the expression $\frac{1}{\left(7\times9\times6\right)^{-6}}$, we will employ exponent rules:

• Step 1: Recognize that the expression has a negative exponent in the denominator. By the rule of negative exponents, we have $\left(a^n\right)^{-m} = \frac{1}{\left(a^n\right)^m} = \left(a^n\right)^m$.
• Step 2: Therefore, the negative exponent $-6$ becomes positive when moved from the denominator: $\left(7 \times 9 \times 6\right)^{-6}$ becomes $\left(7 \times 9 \times 6\right)^{6}$ once moved to the numerator.

Thus, rewriting the expression, we get $\left(7 \times 9 \times 6\right)^6$.

The correct multiple-choice answer is choice 3: $\left(7 \times 9 \times 6\right)^6$.

To solve the problem, follow these steps:

• Given the expression $\frac{1}{8^{-7} \times 9^{-7} \times 5^{-7}}$.
• Apply the negative exponent rule: Each term in the denominator is raised to a negative power.
• Write each term with positive exponents using the reciprocal rule: $\frac{1}{a^{-n}} = a^n$.
• This results in the expression: $8^7 \times 9^7 \times 5^7$.
• The power of a product rule tells us that this is equivalent to $(8 \times 9 \times 5)^7$.

Therefore, the solution is $\left(8 \times 9 \times 5\right)^7$.

Comparing with the multiple-choice options provided, the correct choice is: $\left(8 \times 9 \times 5\right)^7$.