Power of a Product: Variables in the exponent of the power

Let's solve the problem by applying the power of a product rule.

• Step 1: Recognize the product within the exponent. The original expression is $(8 \times 2)^x$.
• Step 2: Apply the power of a product formula, which is $(a \times b)^n = a^n \times b^n$.

By applying this formula, we get:

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

Therefore, the expanded form of $(8 \times 2)^x$ is $8^x \times 2^x$.

Thus, the solution to this problem is $8^x \times 2^x$.

To simplify the expression $(3 \times 6 \times 4)^{2a}$, we apply the power of a product rule:

$(3 \times 6 \times 4)^{2a} = 3^{2a} \times 6^{2a} \times 4^{2a}$

This expression tells us that the exponent $2a$ is distributed to each factor inside the parentheses.

Therefore, the correct answer is $3^{2a} \times 6^{2a} \times 4^{2a}$, corresponding to choice C.

To solve the problem of expressing $(7 \times 5 \times 2)^y$ in another form, we'll use the power of a product rule step by step:

Step 1: Identify and Understand the Given Expression
We are given $(7 \times 5 \times 2)^y$. This indicates that the entire product inside the parentheses is raised to the power $y$.

Step 2: Apply the Power of a Product Rule
The power of a product rule states that $\left(a \times b \times c\right)^n = a^n \times b^n \times c^n$. According to this law, we can distribute the exponent $y$ to each factor within the parentheses.

Step 3: Rewrite the Expression
Applying this rule, we rewrite the expression as:

• Raise the first factor, 7, to the power of $y$: $7^y$.
• Raise the second factor, 5, to the power of $y$: $5^y$.
• Raise the third factor, 2, to the power of $y$: $2^y$.

Putting it all together, the expression becomes $7^y \times 5^y \times 2^y$.

Conclusion
Therefore, the expression $(7 \times 5 \times 2)^y$ is correctly rewritten as $7^y \times 5^y \times 2^y$.

Hence, the corresponding expression is $7^y \times 5^y \times 2^y$.

To solve this problem, let's apply the rule for the power of a product, which states that when a product is raised to a power, each factor is raised to that power.

• Step 1: Write down the expression: $(9 \times 7 \times 4)^a$.
• Step 2: Apply the power of a product rule: $(x \times y \times z)^n = x^n \times y^n \times z^n$.
• Step 3: Raise each factor in the expression to the power $a$.
• Step 4: Simplify the individual expressions: $9^a \times 7^a \times 4^a$.

Thus, the expression $(9 \times 7 \times 4)^a$ simplifies to $9^a \times 7^a \times 4^a$.

The correct answer is $9^a \times 7^a \times 4^a$, which corresponds to choice 2.

To solve this problem, we'll apply the power of a product rule for exponents:

• Identify that $(9 \times 2)^{3x}$ is a product of two numbers $9$ and $2$ raised to the same power $3x$.
• Apply the power of a product rule: $(ab)^n = a^n \times b^n$.
• In our case, $a = 9$, $b = 2$, and $n = 3x$.
• Thus, $(9 \times 2)^{3x} = 9^{3x} \times 2^{3x}$.

Therefore, the expression $(9 \times 2)^{3x}$ simplifies to $9^{3x} \times 2^{3x}$.

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

• Step 1: Identify each factor inside the parentheses: $6$, $7$, and $10$.
• Step 2: Use the power of a product rule, $(abc)^n = a^n \times b^n \times c^n$, to distribute the exponent $y + 4 + a$ to each factor.
• Step 3: Rewrite the expression with each base raised to the power of $y + 4 + a$.

Now, let's work through each step:
Step 1: We have $6 \times 7 \times 10$ as the base of the expression inside the parentheses.
Step 2: According to the power of a product rule, we distribute $y + 4 + a$ across each base, resulting in $6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a}$.
Step 3: Therefore, the expression $(6 \times 7 \times 10)^{y+4+a}$ expands to $6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a}$.

Therefore, the solution to the problem is $6^{y+4+a}\times7^{y+4+a}\times10^{y+4+a}$.

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

• Step 1: Identify the current expression and its structure.
• Step 2: Apply the power of a product rule.
• Step 3: Simplify the expression by distributing the exponent to each factor.

Now, let's work through each step:
Step 1: The given expression is $(7 \times 5 \times 2)^{y+4}$. This is a product inside the parentheses raised to a power.
Step 2: According to the power of a product rule, $(a \times b \times c)^n = a^n \times b^n \times c^n$. We can apply this rule here.
Step 3: Applying the rule, the expression becomes $(7)^{y+4} \times (5)^{y+4} \times (2)^{y+4}$.

Therefore, the correctly expanded expression is $7^{y+4}\times5^{y+4}\times2^{y+4}$.

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

• Step 1: Identify the expression as $(2 \times 8)^{2y+2}$.
• Step 2: Apply the "Power of a Product" rule: $(a \times b)^n = a^n \times b^n$.
• Step 3: Apply this to the bases 2 and 8 in the given expression.

Now, let's work through each step:
Step 1: In our problem, the expression $(2 \times 8)^{2y+2}$ needs to be expanded.
Step 2: According to the exponent rule, we can rewrite the expression as $2^{2y+2} \times 8^{2y+2}$.
Step 3: We have applied the power to each individual base within the parentheses.

Therefore, the corresponding expression is $2^{2y+2} \times 8^{2y+2}$.

To solve this problem, we'll follow the steps below:

First, recognize that the expression given is $(7 \times 8)^{b+a}$. We aim to use the exponent rule known as the power of a product, which states that $(xy)^n = x^n \times y^n$.

Applying this rule to the problem at hand, we have:

$(7 \times 8)^{b+a} = 7^{b+a} \times 8^{b+a}$.

Thus, the expression is expanded to show each base (7 and 8) raised to the combined power of $b+a$.

Checking the choices provided, the expanded expression matches option 2.

Therefore, the correct expression is $7^{b+a}\times8^{b+a}$.

To solve this problem, we need to apply the power of a product rule to the given expression, $(5 \times 3)^{6x}$.

According to the exponent rule for the power of a product, when an expression in the form $(a \times b)^n$ is encountered, it can be expanded to $a^n \times b^n$. Here, $a = 5$, $b = 3$, and $n = 6x$.

By applying this rule, we distribute the exponent $6x$ to each of the bases within the parentheses:

• First, apply the exponent to 5, resulting in $5^{6x}$.
• Then, apply the exponent to 3, resulting in $3^{6x}$.

Thus, the expression $(5 \times 3)^{6x}$ simplifies to $5^{6x} \times 3^{6x}$. This shows that the exponent $6x$ is correctly applied to each element within the parentheses.

Therefore, the expression $(5 \times 3)^{6x}$ is equivalent to $5^{6x} \times 3^{6x}$.

To solve this problem, we will apply the power of a product rule.

• Step 1: Identify the given expression $(3 \times 4)^{3x+1}$.

• Step 2: Apply the power of a product rule: $(ab)^n = a^n \times b^n$.

• Step 3: Rewrite the expression using the rule:

By applying $(3 \times 4)^{3x+1} = 3^{3x+1} \times 4^{3x+1}$, we distribute the exponent to each base within the parentheses.

Therefore, the correct expression is $3^{3x+1} \times 4^{3x+1}$.

To solve this problem, we will apply the power of a product rule. This rule states that when we have a product inside a power, such as $(a \times b)^n$, it can be rewritten as $a^n \times b^n$. Here, our expression is $(4 \times 6)^{b+2}$.

Let's apply this rule step by step:

• Identify the base of the expression: $4 \times 6$.
• The exponent applied to this base is $b + 2$.
• Apply the power of a product rule: $(4 \times 6)^{b+2} = 4^{b+2} \times 6^{b+2}$.

By distributing the exponent $b + 2$ to each factor of the product, we successfully rewrite the expression using the laws of exponents. The rewritten expression is $4^{b+2} \times 6^{b+2}$.

Therefore, the final answer to the problem is $4^{b+2} \times 6^{b+2}$.

To solve this problem, we will follow these steps:

• Identify the given expression: $(4 \times 6)^b$.
• Apply the power of a product rule: $(a \times b)^n = a^n \times b^n$.
• Separate the expression by distributing the exponent to both elements in the product.

Now, let us apply these steps:

The expression we start with is $(4 \times 6)^b$. According to the power of a product rule, $(a \times b)^n = a^n \times b^n$, we distribute the exponent $b$ to each base:

$(4 \times 6)^b = 4^b \times 6^b$.

Therefore, the expression $(4 \times 6)^b$ can be rewritten as $4^b \times 6^b$, which corresponds to choice 1.

We use the power law for a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

That is, a power applied to a multiplication between parentheses is applied to each term when the parentheses are opened,

We apply it in the problem:

$(4\cdot9\cdot11)^a=4^a\cdot9^a\cdot11^a$

Therefore, the correct answer is option b.

Note:

From the power property formula mentioned, we can understand that it works not only with two terms of the multiplication between parentheses, but also valid with a multiplication between multiple terms in parentheses. As we can see in this problem.

Let's begin by using the distributing exponents rule (An exponent outside of a parentheses needs to be distributed across all the numbers and variables within the parentheses)

$(x\cdot y)^n=x^n\cdot y^n$We first apply this rule to the given problem:

$(2\cdot4\cdot8)^{a+3}= 2^{a+3}4^{a+3}8^{a+3}$When then we apply the power to each of the terms of the product inside the parentheses separately and maintain the multiplication.

The correct answer is option d.

To solve this problem, follow these steps:

• Step 1: Identify the expression provided: $3^x \times 7^x \times 5^x$.

• Step 2: Apply the exponent rule for powers of a product: when multiple terms with the same exponent are multiplied, they can be combined under one power. This gives us: $(3 \times 7 \times 5)^x$.

Therefore, the simplified expression is $\left(3 \times 7 \times 5\right)^x$, which matches our final result.

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

• Step 1: Identify the common exponent in all terms.

• Step 2: Apply the power of a product rule to combine the terms.

Now, let's work through each step:
Step 1: We have the expression $4^a \times 8^a \times 2^a$, where each base is raised to the same power, $a$.
Step 2: Using the power of a product rule, we can combine the terms into a single expression: $(4 \times 8 \times 2)^a$.
Therefore, in terms of the choice list, the corresponding expression is $\left(4\times8\times2\right)^a$, which is choice 2.

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

• Step 1: Identify the expression to simplify, [$6^t \times 9^t$].

• Step 2: Apply the Power of a Product Rule, [$a^m \times b^m = (a \times b)^m$].

Now, let's work through each step:

Step 1: We start with the expression $6^t \times 9^t$.

Step 2: Using the Power of a Product Rule, we rewrite this expression as $(6 \times 9)^t$.

Thus, the correct equivalent expression is $\left(6 \times 9\right)^t$.

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

• Step 1: Identify the common exponent in the expression.
• Step 2: Use the power of a product rule to rewrite the expression with a single exponent.

Now, let's work through each step:

Step 1: Recognize that each term in the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ has the same exponent, which is $2a$.

Step 2: Apply the power of a product rule. This rule states that $(x \times y \times z)^n = x^n \cdot y^n \cdot z^n$, which can be reversed to combine the terms.

With this understanding, the expression $5^{2a} \times 8^{2a} \times 7^{2a}$ can be rewritten as a single exponent expression by combining the bases:

$(5 \cdot 8 \cdot 7)^{2a}$.

Therefore, the solution to this problem is $\left(5 \times 8 \times 7\right)^{2a}$.

To solve this problem, we'll simplify the expression by applying the properties of exponents:

• Step 1: Recognize that both bases, $5$ and $3$, have the same exponent $y$.
• Step 2: Apply the power of a product rule in reverse: for terms $a^y \times b^y$, this simplifies to $(a \times b)^y$.
• Step 3: Replace $a$ with $5$ and $b$ with $3$ to get $(5 \times 3)^y$.

Let's work through the solution with these steps:

Given the expression $5^y \times 3^y$, both terms share the same exponent $y$. Therefore, we can combine them by multiplying the bases and keeping the common exponent:

This simplification follows directly from the rule of exponents, which states $a^n \times b^n = (a \times b)^n$ when $n$ is the same for both terms.

Therefore, the simplified expression is $\left(5 \times 3\right)^y$.