Multiplication of Powers: Variables in the exponent of the power

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of $3$

$3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}$

To reduce the expression $5^{2x} \times 5^x$, we will use the exponent multiplication rule:

When multiplying powers with the same base, add the exponents:
Thus, $5^{2x} \times 5^x = 5^{2x + x}$.

Hence, the correct choice is: $5^{2x + x}$.

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

• Step 1: Confirm the given expression is $4^x \times 4^2 \times 4^a$.
• Step 2: Apply the exponent rule for multiplication of powers: if $b^m \times b^n = b^{m+n}$, use this with base 4.
• Step 3: Add the exponents of each term.

Let's work through these steps:

Step 1: The expression we have is $4^x \times 4^2 \times 4^a$.

Step 2: Since all parts of the product have the same base $4$, we can use the rule for multiplying powers: $4^x \times 4^2 \times 4^a = 4^{x+2+a}$.

Step 3: The simplified expression is obtained by adding the exponents: $x + 2 + a$.

Therefore, the expression $4^x \times 4^2 \times 4^a$ simplifies to $4^{x+2+a}$.

To solve this problem, we'll use the property of exponents for multiplying powers with the same base:

• Step 1: Identify that all terms have the same base, which is $8$. The equation is given as $8^a \times 8^2 \times 8^x$.

• Step 2: Apply the multiplication property of exponents: $b^m \times b^n = b^{m+n}$.

• Step 3: Add the exponents: $(a) + (2) + (x)$ to get the new exponent for the single base.

By applying these steps, we obtain:

$8^{a+2+x}$

This result matches choice 1, confirming that this is the correct simplified expression.

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

• Step 1: Identify the common base
• Step 2: Apply the property of exponents
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The problem gives us the expression $2^a \times 2^2$. Here, the common base is 2.
Step 2: We'll apply the property of exponents, which states that for the same base, you add the exponents: $b^m \times b^n = b^{m+n}$. In this case, it will be $2^{a+2}$.
Step 3: Rewriting the expression using this rule, we get: $2^a \times 2^2 = 2^{a+2}$.

Therefore, the solution to the problem is $2^{a+2}$.

To solve this problem, we'll apply the exponent rule for multiplying powers with the same base.

• Step 1: Identify the base. The base for both terms is 3.
• Step 2: Apply the multiplication of powers rule. According to the rule, when multiplying powers with the same base, we add their exponents: $3^4 \times 3^x = 3^{4+x}$.
• Step 3: Write down the simplified form of the expression. The simplified expression of $3^4 \times 3^x$ is: $3^{4+x}$

Therefore, the solution to the expression $3^4 \times 3^x$ simplifies to $3^{4+x}$.

Hence, the correct choice is $3^{4+x}$, matching answer choice 1.

To solve this problem, we will follow these steps:

• Step 1: Identify the given expression and the operation.

• Step 2: Apply the rule for multiplying powers with the same base.

• Step 3: Simplify the expression to reach the final answer.

Let's proceed with the solution:
Step 1: We are given the expression $4^x \times 4^x$. Both terms have the same base of 4 and an exponent of $x$.

Step 2: According to the exponent multiplication rule, $a^m \times a^n = a^{m+n}$. Here, since both bases are 4, we can simplify using this rule.

Step 3: We add the exponents, giving us:

$4^x \times 4^x = 4^{x+x} = 4^{2x}$

Therefore, the correct solution to the problem is $4^{x+x}$.

To solve the problem $7^{x+1}\times7^x$, follow these steps:

Step 1: Use the rule for multiplying powers with the same base:

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

Here, the base $a$ is $7$, and the exponents are $x+1$ and $x$.

Step 2: Add the exponents:

The expression becomes $7^{(x+1) + x}$.

Step 3: Simplify the exponents:

$(x+1) + x = 2x + 1$

Step 4: Write the final expression:

The simplified expression is $7^{2x+1}$.

Therefore, our solution matches choice 3.

The solution to the problem is $7^{2x+1}$.

To solve the expression $5^2 \times 5^a \times 5^3$, we will make use of the exponent rule for multiplication, which states that if you multiply powers with the same base, you add the exponents:

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

Let's apply this rule step by step:

• Step 1: Identify the base of the power terms. In this problem, the base is $5$.
• Step 2: Write down all the exponents. The exponents we have are $2$, $a$, and $3$.
• Step 3: Add the exponents together:
  $2 + a + 3$.
• Step 4: Simplify the sum: $2 + a + 3 = 5 + a$.
• Step 5: Express the final result as a single power:
  The expression can be rewritten using the exponent rule as $5^{5+a}$.

Thus, the final simplified expression is $5^{5+a}$.

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

• Step 1: Identify the exponents in the given expression.
• Step 2: Use the property of exponents for multiplication by like bases.
• Step 3: Add the exponents together and simplify.

Now, let's work through each step:

Step 1: The original expression is $10^{a+b} \times 10^{a+1} \times 10^{b+1}$.

Step 2: Since the base (10) is the same for all terms, we add the exponents:

Step 3: Simplifying further:

Thus, the expression simplifies to:

Therefore, the solution to the problem is $10^{2b+2a+2}$.

We'll use the law of exponents for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$
Note that this law applies to any number of terms being multiplied, not just two terms. For example, when multiplying three terms with the same base, we get:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$
When we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication and so on...

Let's return to the problem:

Notice that all terms in the multiplication have the same base, so we'll use the above law:

$2^{2x+1}\cdot2^5\cdot2^{3x}=2^{2x+1+5+3x}=2^{5x+6}$

$$Therefore, the correct answer is a.

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$
We apply the property for this problem:

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6= 4^{2y+(-5)+(-y)+6}=4^{2y-5-y+6}$
We simplify the expression we got in the last step:

$4^{2y-5-y+6} =4^{y+1}$
When we add similar terms in the exponent.

Therefore, the correct answer is option c.

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply the property to our expression:

$7^{2x+1}\cdot7^{-1}\cdot7^x=7^{2x+1+(-1)+x}=7^{2x+1-1+x}$We simplify the expression we got in the last step:

$7^{2x+1-1+x}=7^{3x}$When we add similar terms in the exponent.

Therefore, the correct answer is option d.

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 will use the rule of exponents that allows us to expand the sum $a + b + c$ in the exponent:

• Given: $4^{a+b+c}$
• According to the exponent rule $x^{m+n+p} = x^m \times x^n \times x^p$, we can express:
• Step: Break down $4^{a+b+c}$ to:
• $4^a \times 4^b \times 4^c$

Therefore, the expanded form of the equation is $4^a \times 4^b \times 4^c$.

To solve this problem, we will use the properties of exponents:

• Step 1: Recognize that the exponent $10a + 5x$ can be expressed as the sum of two terms, namely $(5a + 5x) + 5a$.
• Step 2: Apply the property $g^{m+n} = g^{m} \times g^{n}$ to the expression $g^{10a + 5x}$.
• Step 3: Expand the expression using the identified split:
  Since $10a + 5x = (5a + 5x) + 5a$, we have:
  $g^{10a + 5x} = g^{(5a + 5x) + 5a} = g^{5a + 5x} \times g^{5a}$.

Thus, the expanded form of the expression is given by $g^{5a+5x} \times g^{5a}$.

To tackle this problem, we need to expand the expression $7^{2x+7}$ using the properties of exponents:

• Step 1: Identify the structure of the exponent in the form $2x + 7$.
• Step 2: Recognize this as a sum of two components: $2x$ and $7$.
• Step 3: Apply the exponent rule $a^{m+n} = a^m \times a^n$ to separate the expression into two distinct exponent terms.

Now, let's proceed with the solution:

Step 1: Given the exponent is $2x + 7$, break it down into individual components:
$2x + 7 = x + x + 7$.

Step 2: Applying the rule $a^{m+n} = a^m \times a^n$:
$7^{2x+7} = 7^{x+x+7} = 7^x \times 7^{x+7}$.

Therefore, the expanded form of the expression is $7^x \times 7^{x+7}$.

To solve the problem, we can follow these steps:

• Step 1: Recognize that the given expression is $2^{2a+a}$.
• Step 2: Use the Power of a Power Rule for exponents, which allows us to write $a^{m+n} = a^m \times a^n$.
• Step 3: Rewrite the expression as follows:

Given: $2^{2a+a}$

Step 4: Simplify the exponent by splitting it:

Since the expression in the exponent is $2a+a$, we can write:

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

Thus, applying the properties of exponents correctly leads us to the expanded form.

Therefore, the expanded equation is $2^{2a} \times 2^a$.

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

• Step 1: Analyze the given expression's exponent.
• Step 2: Apply the exponent addition rule to expand the expression.
• Step 3: Identify the correct choice from a set of given options.

Now, let's work through each step:
Step 1: The expression given is $3^{2a + x + a}$. Here the exponent is $2a + x + a$.
Step 2: We apply the rule $b^{m+n} = b^m \times b^n$ by rewriting the exponent sum as individual terms: $(2a)$, $x$, and $a$.
Thus, we can rewrite the expression using the property of exponents: $3^{2a + x + a} = 3^{2a} \times 3^x \times 3^a$.
Step 3: Upon expanding, the solution corresponds to option :

$3^{2a}\times3^x\times3^a$

Therefore, the expanded expression is $3^{2a}\times3^x\times3^a$.

Note that there is multiplication operation between all terms in the expression, hence we'll first apply the distributive property of multiplication in order to handle the coefficients of terms raised to powers, and the terms themselves separately. For greater clarity, let's break this down into steps:

$a^x\cdot3a^y\cdot a^2\cdot2a=3\cdot2\cdot a^x\cdot a^y\cdot a^2\cdot a=6\cdot a^x\cdot a^y\cdot a^2\cdot a$

Due to the multiplication operation between all terms we could do this, it should be noted that we can (and it's preferable to) skip the middle step, meaning:

Write directly:$a^x\cdot3a^y\cdot a^2\cdot2a=6\cdot a^x\cdot a^y\cdot a^2\cdot a$

From here on we won't write the multiplication sign anymore instead we simply place the terms next to each other\ place the term next to its coefficient to indicate multiplication between them,

Proceed to apply the law of exponents for multiplication of terms with identical bases:

$c^m\cdot c^n=c^{m+n}$Note also that this law applies to any number of terms being multiplied and not just two, for example for three terms with identical bases we obtain:

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

Whilst we used law of exponents twice, we can also perform the same calculation for four terms or 5 and so on..,

Let's return to the problem, and apply the above law of exponents:

$6a^xa^yaa^2=6a^{x+y+2+1}=6a^{x+y+3}$

Therefore the correct answer is d.

Important note:

Here we need to emphasize that we should always ask the question - what is the exponent being applied to?

For example, in this problem the exponent applies only to the bases of-

$a$$$and not to the numbers, more clearly, in the following expression: $5c^7$the exponent applies only to $c$ and not to the number 5,

whereas when we write:$(5c)^7$the exponent applies to each term of the multiplication inside the parentheses,

meaning:$(5c)^7=5^7c^7$

This is actually the application of the law of exponents:

$(w\cdot r)^n=w^n\cdot r^n$

which follows both from the meaning of parentheses and from the definition of exponents.