Multiplication of Powers: Variable in the exponent of the power

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

• Identify the given expression and its components.

• Recognize that both terms share the same base.

• Apply the rule for multiplying powers with the same base, $b^m \times b^n = b^{m+n}$.

• Calculate the exponent by adding the powers together.

Let's work through these steps:

Given the expression:
$6^a \times 6^{4a}$

Since both terms share the same base (6), we use the rule for multiplying powers with the same base:

$6^a \times 6^{4a} = 6^{a + 4a}$.

Simplify the exponent:

$a + 4a = 5a$.

Thus, the expression simplifies to:

$6^{5a}$.

Since the correct placement of our steps has directly given us choice C and corroborated choice B as intermediary work, B+C represents the comprehensive workings of the problem; thus, B+C are correct is the most comprehensive solution.

To solve this problem, we'll simplify the given equation using the rules of exponents:

• Identify that all terms in the product $6^x \times 6^a \times 6^b$ have the same base, which is 6.
• Apply the exponent multiplication rule: When multiplying powers with the same base, we add the exponents together. Therefore, the expression becomes $6^{x+a+b}$.

By applying this exponent rule, we determine that the simplified expression is $6^{x+a+b}$.

Therefore, the solution to the problem is $6^{x+a+b}$.

To solve this problem, we'll start by applying the rules for multiplying powers with the same base.

• Step 1: Identify the expression given as $7^{x+a} \times 7^a \times 7^x$.
• Step 2: Apply the rule $a^m \times a^n = a^{m+n}$ to combine the exponents since all terms have the same base, 7.

Let's proceed to simplify:

Combine the exponents:

$7^{x+a} \times 7^a \times 7^x = 7^{(x+a) + a + x}$.

Now, simplify the addition of the exponents:

$7^{(x+a) + a + x} = 7^{x + a + a + x}$.

Combine like terms in the exponent:

$x + a + a + x = 2x + 2a$.

Thus, the expression simplifies to:

$7^{2x+2a}$.

Therefore, the simplification of the given expression is $7^{2x+2a}$.

To solve this problem, let's simplify the expression $8^{3x} \times 8^{3y} \times 8^{2y+x}$ using exponent rules:

Step 1: Identify the exponents in each term:
- The first term is $8^{3x}$ with an exponent of $3x$.
- The second term is $8^{3y}$ with an exponent of $3y$.
- The third term is $8^{2y+x}$ with an exponent of $2y+x$.

Step 2: Apply the multiplication of powers rule:
Since all terms have the same base of 8, add the exponents: $(3x) + (3y) + (2y + x)$.

Step 3: Simplify the expression:
Adding the terms in the exponent gives us: $3x + 3y + 2y + x = 4x + 5y$.

Therefore, the simplified expression is $8^{4x+5y}$.

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

• Step 1: Recognize that all the terms have the same base, which is 9.
• Step 2: Apply the exponent addition rule to combine the exponents.
• Step 3: Simplify the result by performing algebraic addition on the exponents.

Now, let's work through each step:
Step 1: The problem gives us the expression $9^{2a} \times 9^{2x} \times 9^a$. All terms share the same base, which is 9.
Step 2: Using the property of exponents $b^m \times b^n = b^{m+n}$, add the exponents: $(2a) + (2x) + a$.
Step 3: Combine like terms: $2a + a + 2x = 3a + 2x$. So, the expression becomes $9^{3a+2x}$.

Therefore, the simplified form of the given equation is $9^{3a+2x}$.

We use the law of exponents to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the law to given the problem:

$7^x\cdot7^{-x}=7^{x+(-x)}=7^{x-x}=7^0$In the first stage we apply the above power rule and in the following stages we simplify the expression obtained in the exponent,

Subsequently, we use the zero power rule:

$X^0=1$We obtain:

$7^0=1$Lastly we summarize the solution to the problem.

$7^x\cdot7^{-x}=7^{x-x}=7^0 =1$Therefore, the correct answer is option B.

Begin by applying the distributive property of multiplication and proceed to arrange the algebraic expression according to like bases:

$a^ya^xa^{6\text{ }}b^97^y$

Next, we'll use the power rule to multiply terms with the same base:

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

Note that this rule applies to any number of terms in multiplication, not just two. For example, when multiplying three terms with the same base, we obtain the following:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$

Therefore, we can combine all terms with the same base under one base:

$a^{y+x+6}b^97^y$

Note that we could only combine terms with identical bases using this rule,

From here we can see that the expression cannot be simplified further, and therefore this is the correct answer, which is answer C (since the distributive property of multiplication holds).

Begin by applying the distributive property of multiplication in order to arrange the algebraic expression according to like terms:

$x^3x^4x^{3+y}y^4z^6$

$$Next, we'll use the law of exponents to multiply terms with the same base:

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

Note that this law applies to any number of terms being multiplied, not just two. For example, when multiplying three terms with the same base, we obtain the following:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$

Therefore, we can combine all terms with the same base under one base:

$x^{3+4+3+y}y^4z^6=x^{10+y}y^4z^6$

In the second step we simply added the exponents together.

Note that we could only combine terms with the same base using this law,

From here we can see that the expression cannot be simplified further, and therefore this is the correct and final answer which is answer D.

To solve this problem, we need to simplify the expression $8^{-3x} \times 8^{-3y} \times 8^{2y+x}$ using exponent rules. Let's break it down step-by-step.

First, recognize that each term in the product has the same base, which is 8. The multiplication rule for exponents allows us to add the exponents when multiplying like bases. Our expression is:

$8^{-3x} \times 8^{-3y} \times 8^{2y+x}$

According to the rule $a^m \times a^n = a^{m+n}$, we add the exponents of each term:

$(-3x) + (-3y) + (2y + x)$

Combine the exponents inside the parentheses:

$-3x - 3y + 2y + x$

Now, group and simplify like terms:

• The terms with $x$: $-3x + x = -2x$
• The terms with $y$: $-3y + 2y = -y$

Combine these results:

The expression becomes $-2x - y$.

Therefore, the reduced form of the original expression is:

$8^{-2x-y}$

The given expression is $a^{-3} \times a^5 \times a^{-4}$.

To simplify, use the product of powers property, which states that when multiplying like bases, you add the exponents:

• Apply the rule: $a^{-3} \times a^5 \times a^{-4} = a^{-3+5-4}$.
• Calculate the sum of the exponents: $-3 + 5 - 4 = -2$.

This simplifies the expression to $a^{-2}$.

Note that $a^{-2}$ can also be expressed as $\frac{1}{a^2}$ using the property of negative exponents $(a^{-n} = \frac{1}{a^n})$.

Now, let's evaluate the choices:

• Choice 1: $a^{-2}$, which matches our simplification.
• Choice 2: $\frac{1}{a^2}$, which is another form of $a^{-2}$.
• Choice 3: $a^{-3+5-4}$, which correctly reflects the intermediate step we computed.
• Choice 4 states "All answers are correct," which, considering all interpretations and simplifications are valid, is indeed true.

Therefore, the correct answer is: All answers are correct, as each choice corresponds to a logical step or equivalent expression in reducing the equation.