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

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 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'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 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 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 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.

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'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 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 $7^y \times 5^y \times 2^y$.

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 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 $9^{3x} \times 2^{3x}$.

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.

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 use the "Power of a Product" rule.

We begin with the expression $5^{y+1} \times 3^{y+1}$.

Notice that both terms share the same exponent, $y+1$.

According to the Power of a Product rule, $a^m \times b^m = (a \times b)^m$. This means we can combine $5^{y+1} \times 3^{y+1}$ into a single expression.

Let's apply the formula:

• Identify each base: $a = 5$ and $b = 3$.

• The shared exponent is $m = y+1$.

• Substitute into the formula: $(5 \times 3)^{y+1}$.

Therefore, the corresponding expression is $\left(5 \times 3\right)^{y+1}$.

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 $\left(5 \times 3\right)^y$.

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 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 $\left(6 \times 9\right)^t$.