Power of a Product: Variable in the base of the power

We use the formula:

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

$(5\times x\times3)^3=(15x)^3$

$(15x)^3=(15\times x)^3$

$15^3x^3$

We use the formula:

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

$(y\times x\times3)^5=y^5x^53^5$

We use the formula

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$a^2b^28^2$

We use the formula:

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$(a\times5\times6\times y)^5=(a\times30\times y)^5$

$a^530^5y^5$

Step 1: The problem provides the expression $\left(c \times b \times a\right)^2$ and asks us to expand it.
Step 2: We'll use the exponent rule for the power of a product, which states that $(xyz)^n = x^n \times y^n \times z^n$. Applying this rule to our expression, we get:
$\left(c \times b \times a\right)^2 = c^2 \times b^2 \times a^2$
Since multiplication is commutative, the order of the factors doesn't affect the product. Therefore, the expression can also be written as:
$\left(c \times b \times a\right)^2 = a^2 \times b^2 \times c^2$

Therefore, the correct expressions are $c^2 \times b^2 \times a^2$ and $a^2 \times b^2 \times c^2$.

The problem requires us to simplify $(6 \times b)^4$.

• Step 1: Recognize that $(6 \times b)$ is a product of two factors $6$ and $b$.
• Step 2: Apply the Power of a Product Rule, which states that $(ab)^n = a^n \times b^n$.

Let's apply this rule to $(6 \times b)^4$:
Using the rule, we distribute the exponent $4$ to each component of the product:

Thus, the expression simplifies to $6^4 \times b^4$.

Therefore, the simplified expression is:

$6^4 \times b^4$.

To expand the expression $\left(b \times 9 \times a\right)^6$, we'll apply the power of a product rule for exponents, which states that $(xyz)^n = x^n \times y^n \times z^n$.

• Step 1: Identify each factor within the base: $b$, $9$, and $a$.
• Step 2: Apply the exponent to each factor individually:
• $b^6$
• $9^6$
• $a^6$

By multiplying these results together, the expanded form is $b^6 \times 9^6 \times a^6$.

Therefore, the correct expression is $b^6 \times 9^6 \times a^6$.

To solve the expression $(2 \times x)^2$, we'll follow these steps:

• Step 1: Identify the base of the exponent, which is the product $2 \times x$.
• Step 2: Apply the Power of a Product rule, which states that $(ab)^n = a^n \times b^n$.
• Step 3: Distribute the exponent to both factors within the parentheses:

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

Calculating $2^2$ gives:

$2^2 = 4$

So, the expression simplifies to:

$4 \times x^2$

Therefore, the correct expression is $2^2 \times x^2$.

This corresponds to Choice 2.

To solve the expression $\left(5 \times b \times a\right)^4$, we'll apply the Power of a Product Rule, which states that when a product is raised to an exponent, each factor in the product is raised to that exponent individually.

• First, identify each factor in the product: $5$, $b$, and $a$.

• Next, apply the exponent to each factor:

  • The number $5$ becomes $5^4$.

  • The variable $b$ becomes $b^4$.

  • The variable $a$ becomes $a^4$.

• Finally, multiply these results together to obtain the simplified expression.

Therefore, the expression $\left(5 \times b \times a\right)^4$ simplifies to $5^4 \times b^4 \times a^4$, which corresponds to Choice 3.

To solve the problem $(a \times 3)^3$, we'll apply the power of a product rule which states that $(x \cdot y)^n = x^n \cdot y^n$.

Step 1: Identify the individual factors within the parentheses. In this expression, $a$ and 3 are multiplied together and are being raised to the power of 3.

Step 2: Apply the power of a product property: Distribute the exponent of 3 to both $a$ and 3 inside the parentheses. We do so as follows:
$(a \times 3)^3 = a^3 \times 3^3$

Step 3: Express the result clearly. The expression simplifies to:
$a^3 \times 3^3$.

Therefore, the correct answer to this problem is $a^3 \times 3^3$.

To solve the problem $\left(a \times b\right)^3$, we'll apply the rules of exponents, specifically the Power of a Product rule.

• Step 1: Understand the Power of a Product Rule
  The Power of a Product rule states that when you raise a product to an exponent, you can apply the exponent to each factor inside the parentheses individually. Mathematically, this is expressed as: $\left(a \times b\right)^n = a^n \times b^n$

• Step 2: Apply the Rule to the Given Expression
  Given the expression $\left(a \times b\right)^3$, we can apply the Power of a Product rule by raising each factor inside the parentheses to the power of 3: $\left(a \times b\right)^3 = a^3 \times b^3$

• Step 3: Simplify the Expression
  After applying the exponent to both $a$ and $b$, the expression simplifies to: $a^3 \times b^3$

Therefore, the corresponding expression for $\left(a \times b\right)^3$ is $a^3 \times b^3$.

To solve the problem, we will use the rule of exponents known as the power of a product rule, which states that for any real numbers or expressions $x$, $y$ raised to a power $n$, the following holds:

$(x \times y)^n = x^n \times y^n$.

We have the expression $\left(b \times z \times a\right)^5$. According to the power of a product rule, we apply the exponent 5 to each factor inside the parenthesis.

Let's break it down:

• Apply $5$ to $b$: $(b)^5 = b^5$.
• Apply $5$ to $z$: $(z)^5 = z^5$.
• Apply $5$ to $a$: $(a)^5 = a^5$.

By applying the exponent to each factor, we obtain:
$(b \times z \times a)^5 = b^5 \times z^5 \times a^5$.

Since multiplication is commutative, we can write it in any order, and a common convention is ordering it alphabetically:

Thus, $a^5 \times b^5 \times z^5$ is the simplified expression.

Therefore, the correct answer to the problem is $a^5 \times b^5 \times z^5$, which corresponds to choice 1.

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

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:
Step 1: The problem provides the expression $\left(y \times a\right)^5$ and asks us to write the corresponding expanded expression.
Step 2: We'll use the exponent rule for the power of a product, which states that $(xy)^n = x^n \times y^n$.
Step 3: Applying this rule, we raise each factor inside the parentheses to the fifth power: $y^5 \times a^5$.

Therefore, the solution to the problem is $y^5 \times a^5$.

Among the given choices, Choice 1: $y^5 \times a^5$ is the correct expression.

To solve this problem, we will apply the power of a product rule to the expression $(y \times x)^2$.

The power of a product rule states:

• $(a \times b)^n = a^n \times b^n$

In this case, the product is $y \times x$, and the power is 2. Applying the power of a product rule gives us:

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

Therefore, the expanded form of the given expression is $\mathbf{y^2 \times x^2}$.

To solve this problem, we will apply the power of a product rule to the given expression $\left(y \times 3\right)^2$.

Let's go through the solution step-by-step:

• Step 1: Understand the expression
  The expression $\left(y \times 3\right)^2$ indicates that the product of $y$ and $3$ is squared. This means we need to apply the square to both terms inside the parentheses.

• Step 2: Apply the power of a product rule
  According to the power of a product rule: $(a \times b)^n = a^n \times b^n$. In our case, $a$ is $y$, $b$ is $3$, and $n$ is $2$. Thus, we have: $(y \times 3)^2 = y^2 \times 3^2$.

Therefore, the correct answer to this problem is $y^2 \times 3^2$, which matches choice 4.

We begin by applying the power of a product rule to the expression $(7 \times 4 \times a)^5$.

The power of a product rule states that $(x \times y \times z)^n = x^n \times y^n \times z^n$.

In this case, the expression inside the parentheses is $7 \times 4 \times a$, and it is being raised to the 5th power. We apply the exponent to each factor:

$7^5$

$4^5$

$a^5$

Therefore, the corresponding expression is $\boxed{7^5 \times 4^5 \times a^5}$.

We use the power law for multiplication within parentheses:

$(x\cdot y)^n=x^n\cdot y^n$

We apply it in the problem:

$(y\cdot7\cdot3)^4=y^4\cdot7^4\cdot3^4$

Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we can understand that it applies not only to two terms within parentheses, but also for multiple terms within parentheses.

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 problem, let's use the power of a product rule, which states that the product of several terms raised to the same power can be expressed as one parenthesis where the terms are multiplied, raised to that power.

Given the expression:

• $4^3 \times x^3 \times 6^3$

We observe that all the terms $4$, $x$, and $6$ are each raised to the power of $3$. Therefore, we can represent this expression using a single power as follows:

$\left(4 \times x \times 6\right)^3$

This transformation uses the formula $(a \times b \times c)^n = a^n \times b^n \times c^n$, taking advantage of corresponding exponents.

Therefore, the simplified expression is $\left(4 \times x \times 6\right)^3$.

To solve the problem $3^2 \times y^2 =$, we will make use of the power of a product rule, which helps simplify products with the same exponents.

The rule states that for any numbers $a$ and $b$ and any exponent $m$, $(a^m \times b^m) = (a \times b)^m$.

Using this rule, we can combine $3^2$ and $y^2$ into a single expression:

• Identify the bases: We have a base of 3 and a base of $y$, both raised to the power of 2.

• Apply the formula, combining the bases under a common exponent: $(3 \times y)^2$.

This shows that $3^2 \times y^2 = \left(3 \times y\right)^2$.