Applying Combined Exponents Rules: Applying the formula

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 $3^6$.

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{3^5}{3^2} = 3^{5-2}$

Simplifying, we get $3^3$

Using the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Here, we have $\frac{5^6}{5^4} = 5^{6-4}$. Simplifying, we get $5^2$.

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

• Identify the given information and relevant exponent rules.

• Apply the quotient property of exponents.

• Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression $\frac{6^7}{6^4}$. The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ In this case, $a = 6$, $m = 7$, and $n = 4$.
Step 3: Applying this rule gives us: $\frac{6^7}{6^4} = 6^{7 - 4} = 6^3$

Therefore, the solution to the problem is $6^3$.

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

• Step 1: Identify the provided expression: $(9^2)^4$.

• Step 2: Apply the power of a power rule for exponents.

• Step 3: Simplify by multiplying the exponents.

Now, let's work through each step:

Step 1: We have the expression $(9^2)^4$.

Step 2: Using the power of a power rule ($(a^m)^n = a^{m \cdot n}$), apply it to the expression:

$(9^2)^4 = 9^{2 \times 4}$

Step 3: Simplify by calculating the product of the exponents:

$2 \times 4 = 8$

Therefore, $(9^2)^4 = 9^8$.

The correct expression corresponding to the given problem is $\boxed{9^8}$.

To solve this problem, we need to simplify the expression $\frac{b^5}{b^2}$ using the rules of exponents.

• Step 1: Identify the rule to apply: For any positive integer exponents $m$ and $n$, the rule $\frac{a^m}{a^n} = a^{m-n}$ applies when dividing terms with the same base. In this expression, our base is $b$.

• Step 2: Apply the rule: Substitute the given exponents into the formula: $\frac{b^5}{b^2} = b^{5-2}$

• Step 3: Perform the subtraction: Calculate the exponent $5 - 2$: $b^{5-2} = b^3$

Therefore, the solution to the expression $\frac{b^5}{b^2}$ is $b^3$.

To solve the given expression $\frac{x^6}{x^4}$, we will follow these steps:

• Step 1: Apply the quotient rule for exponents
• Step 2: Simplify the expression
• Step 3: Verify by comparing with the answer choices

Now, let's work through each step:

Step 1: Apply the quotient rule for exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ when dividing powers with the same base.

Step 2: We have $\frac{x^6}{x^4}$. According to the rule:

$\frac{x^6}{x^4} = x^{6-4} = x^2$

Step 3: Verify by comparing with the answer choices:

• Choice 1: $x^{-2}$ – Incorrect as it implies the exponents were added incorrectly.
• Choice 2: $x^2$ – This matches our result.
• Choice 3: $x^{10}$ – Incorrect as it implies the exponents were added instead of subtracted.
• Choice 4: $x^{\frac{2}{3}}$ – Incorrect as it does not match the calculation based on integer exponents.

Therefore, the correct choice is $x^2$, which is Choice 2.

To simplify the expression $(x^3)^4$, we'll follow these steps:

• Step 1: Identify the expression: $(x^3)^4$.
• Step 2: Apply the formula for a power raised to another power.
• Step 3: Calculate the product of the exponents.

Now, let's work through each step:

Step 1: We have the expression $(x^3)^4$, which involves a power raised to another power.

Step 2: We apply the exponent rule $(a^m)^n = a^{m \cdot n}$ here with $a = x$, $m = 3$, and $n = 4$.

Step 3: Multiply the exponents: $3 \times 4 = 12$. This gives us a new exponent for the base $x$.

Therefore, $(x^3)^4 = x^{12}$.

Consequently, the correct answer choice is: $x^{12}$ from the options provided. The other options $x^6$, $x^1$, and $x^7$ do not reflect the correct application of the exponent multiplication rule.

To solve the expression $\frac{y^9}{y^3}$, we will apply the rules of exponents, specifically the power of division rule, which states that when you divide like bases, you subtract the exponents.

Here are the steps to arrive at the solution:

• Step 1: Identify and write down the expression: $\frac{y^9}{y^3}$.

• Step 2: Apply the division rule of exponents, which is $\frac{a^m}{a^n} = a^{m-n}$, for any non-zero base $a$.

• Step 3: Using the division rule, subtract the exponent in the denominator from the exponent in the numerator:$y^{9-3}$

• Step 4: Calculate the exponent: $9 - 3 = 6$

• Step 5: Write down the simplified expression:$y^6$

Therefore, the expression $\frac{y^9}{y^3}$ simplifies to $y^6$.

To solve the problem of finding $7^0$, we will follow these steps:

• Step 1: Identify the general rule for exponents with zero.

• Step 2: Apply the rule to the given problem.

• Step 3: Consider the provided answer choices and select the correct one.

Now, let's work through each step:

Step 1: A fundamental rule in exponents is that any non-zero number raised to the power of zero is equal to one. This can be expressed as: $a^0 = 1$ where $a$ is not zero.

Step 2: Apply this rule to the problem: Since we have $7^0$, and $7$ is certainly a non-zero number, the expression evaluates to 1. Therefore, $7^0 = 1$.

Therefore, the solution to the problem is $7^0 = 1$, which corresponds to choice 2.

To solve this problem, let's follow these steps:

• Understand the zero exponent rule.

• Apply this rule to the given expression.

• Identify the correct answer from the given options.

According to the rule of exponents, any non-zero number raised to the power of zero is equal to $1$. This is one of the fundamental properties of exponents.
Now, apply this rule:

Step 1: We are given the expression $(-3)^0$.
Step 2: Here, $-3$ is our base. We apply the zero exponent rule, which tells us that $(-3)^0 = 1$.

Therefore, the value of $(-3)^0$ is $1$.

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.

We use the zero exponent rule.

$X^0=1$We obtain

$112^0=1$Therefore, the correct answer is option C.

To solve the exercise we use the power property:$(a^n)^m=a^{n\cdot m}$

We use the property with our exercise and solve:

$(3^5)^4=3^{5\times4}=3^{20}$

We use the formula:

$(a^n)^m=a^{n\times m}$

Therefore, we obtain:

$6^{2\times13}=6^{26}$

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.

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.

In the exercise of multiplying powers, we will add up all the powers of the same product, in this case the terms a, b

We use the formula:

$a^n\times a^m=a^{n+m}$

We are going to focus on the term a:

$a^3\times a^2=a^{3+2}=a^5$

We are going to focus on the term b:

$b^4\times b^5=b^{4+5}=b^9$

Therefore, the exercise that will be obtained after simplification is:

$a^5\times b^9$

Using the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$It is important to note that this law is only valid for terms with identical bases,

We notice that in the problem there are two types of terms. First, for the sake of order, we will use the substitution property to rearrange the expression so that the two terms with the same base are grouped together. The, we will proceed to solve:

$k^2t^4k^6t^2=k^2k^6t^4t^2$Next, we apply the power property to each different type of term separately,

$k^2k^6t^4t^2=k^{2+6}t^{4+2}=k^8t^6$We apply the property separately - for the terms whose bases are$k$and for the terms whose bases are$t$We add the powers in the exponent when we multiply all the terms with the same base.

The correct answer then is option b.

We use the power law for multiplication within parentheses:

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

We apply it to the problem:

$(3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4$

Therefore, the correct answer is option b.

Note:

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