Power of a Product Practice Problems - Master Exponent Rules

Step 1: We start with the expression $\left(20 \times 5\right)^7$.
Step 2: We'll apply the power of a product rule, which states $(a \times b)^n = a^n \times b^n$. This gives us: $\left(20 \times 5\right)^7 = 20^7 \times 5^7$ .
Step 3: To verify, notice that both $20^7 \times 5^7$ and $5^7 \times 20^7$ involve the same expression due to the commutative property of multiplication. Also, we can rewrite $\left(20 \times 5\right)$ as $100$, leading to another form: $\left(20 \times 5\right)^7 = 100^7$
Thus, both $20^7 \times 5^7$, $5^7 \times 20^7$, and $100^7$ are equivalent expressions for $\left(20 \times 5\right)^7$.

Therefore, the correct answer choice is (d) "All answers are correct."

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 $13^6 \times 4^6$.

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 given expression is $\left(2\times3\right)^2$. We need to apply the rule of exponents known as the "Power of a Product." This rule states that when you have a product raised to an exponent, you can apply the exponent to each factor in the product individually. Mathematically, this is expressed as: $(a \times b)^n = a^n \times b^n$.

In this case, the expression $\left(2\times3\right)^2$ follows this rule with $a = 2$ and $b = 3$, and $n = 2$.

• First, apply the exponent to the first factor: $2^2$.
• Next, apply the exponent to the second factor: $3^2$.

Therefore, by applying the "Power of a Product" rule, the expression becomes: $2^2 \times 3^2$.

The problem requires us to simplify the expression $(5 \times 7)^3$ using the power of a product rule.

The power of a product rule states that for any numbers $a$ and $b$, and any integer $n$, the expression $(a \times b)^n$ can be expanded to $a^n \times b^n$.

Applying this rule to the given expression:

• Identify the values of $a$ and $b$ as $5$ and $7$, respectively.

• Identify $n$ as $3$.

• Substitute using the rule:
  $(5 \times 7)^3 = 5^3 \times 7^3$

The simplified expression is therefore $5^3 \times 7^3$.