Power of a Product: Inverse formula

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 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 $(3 \times 4)^4$.

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:

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