Combined Exponent Rules Practice Problems & Solutions

In order to solve this problem, we'll follow these steps:

• Step 1: Identify the base and exponents

• Step 2: Use the formula for multiplying powers with the same base

• Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is $(3^4) \times (3^2)$. Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
$a^m \times a^n = a^{m+n}$

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
$3^4 \times 3^2 = 3^{4+2} = 3^6$

Therefore, the simplified form of the expression is $3^6$.

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

We apply it in the problem:

$\frac{2^4}{2^3}=2^{4-3}=2^1$

Remember that any number raised to the 1st power is equal to the number itself, meaning that:

$b^1=b$

Therefore, in the problem we obtain:

$2^1=2$

Therefore, the correct answer is option a.

Let's simplify the expression $5^3 \times 2^4 \times 5^2 \times 2^3$ using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

• Step 1: Focus on terms with the same base.
  Combine $5^3$ and $5^2$. Since both terms have the base $5$, we apply the rule $a^m \times a^n = a^{m+n}$: $5^3 \times 5^2 = 5^{3+2} = 5^5$

• Step 2: Combine $2^4$ and $2^3$. Similarly, for the base $2$: $2^4 \times 2^3 = 2^{4+3} = 2^7$

After simplification, the expression becomes:
$5^5 \times 2^7$

Note that in the fraction and its denominator, there are terms with the same base, so we will use the law of exponents for division between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

Let's apply it to the problem:

$\frac{9^9}{9^3}=9^{9-3}=9^6$$$

Therefore, the correct answer is b.

To reduce the expression $a^2 \times a^5 \times a^3$, we will apply the product of powers property of exponents. This property states that when multiplying expressions with the same base, we add their exponents.

• Step 1: Identify the exponents.
  The expression involves the same base $a$ with exponents: 2, 5, and 3.
• Step 2: Add the exponents.
  According to the product of powers property, $a^2 \times a^5 \times a^3 = a^{2+5+3}$.
• Step 3: Simplify the expression.
  Calculate the sum of the exponents: $2 + 5 + 3 = 10$. Therefore, the expression simplifies to $a^{10}$.

Ultimately, the solution to the problem is $a^{10}$. Among the provided choices, is correct: $a^{10}$. The other options $a^5$, $a^8$, and $a^4$ do not correctly reflect the sum of the exponents as calculated.