Power of a Product Practice Problems - Master Exponent Rules

To solve this problem, we'll apply the power of a product rule to the expression $(10 \times 3)^4$.

• Step 1: Identify the expression.
  The given expression is $(10 \times 3)^4$.

• Step 2: Apply the power of a product law.
  According to the rule, $(a \times b)^n = a^n \times b^n$, our expression becomes:
  $(10 \times 3)^4 = 10^4 \times 3^4$.

• Step 3: Evaluate the choices:
  - First choice: $3^4 \times 10^4$
  Rearranging terms, this is equivalent to $10^4 \times 3^4$. Therefore, it matches our transformed expression.
  - Second choice: $30^4$
  Since $30 = 10 \times 3$, $(10 \times 3)^4 = 30^4$. This simplifies to the same expression.
  - Third choice: $10^4 \times 3^4$

  Therefore, the solution is that all answers are correct.

The given expression is $(12 \times 3)^5$. The power of a product rule states that $(a \times b)^n = a^n \times b^n$. We will apply this formula to the expression.

• Firstly, identify the base of the power as the product $12 \times 3$.

• Secondly, recognize that the exponent applied to this product is 5.

• According to the rule, the power of a product can be distributed to each factor in the product, which means: $(12 \times 3)^5 = 12^5 \times 3^5$.

Therefore, the expression $(12 \times 3)^5$ corresponds to $12^5 \times 3^5$.

The expression in question is $\left(13\times4\right)^6$. This expression involves raising a product to a power and requires the application of the power of a product exponent rule, which states:

• When you have a product raised to an exponent, you can distribute the exponent to each factor in the product separately.

Mathematically, any numbers $a$ and $b$ and a positive integer $n$ can be written as$(a \times b)^n = a^n \times b^n$.

Applying this rule to our expression, $\left(13 \times 4\right)^6$ becomes $13^6 \times 4^6$.

Therefore, the expression $\left(13 \times 4\right)^6$ simplifies to $13^6 \times 4^6$.

To solve the expression, we can apply the rule for the power of a product, which states that$\left(a \times b\right)^n = a^n \times b^n$.

In this case, our expression is $\left(2\times11\right)^5$, where$a = 2$ and $b = 11$, and $n = 5$.

Applying the power of a product rule gives us:

• $a^n = 2^5$

• $b^n = 11^5$

Therefore, $\left(2\times11\right)^5 = 2^5 \times 11^5$.

To solve the question, we need to apply the power of a product exponent rule. The formula states that for any real numbers $a$ and $b$, and any integer$n$:

• $(a \times b)^n = a^n \times b^n$

Looking at our expression, we can see that:

• $a = 2$

• $b = 4$

• $n = 10$

Now, if we apply the formula:

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

Therefore, the expression $(2 \times 4)^{10}$ is equivalent to $2^{10} \times 4^{10}$.