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 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 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 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'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 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 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 $10^{2b+2a+2}$.

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 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 $4^{x+x}$.

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

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 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 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 $7^x \times 7^{x+7}$.

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.

First, let’s carefully expand the parentheses by applying two exponent laws.

The first law is the product of powers rule, which applies when an exponent is raised over a product of terms:

$(a\cdot b)^n=a^n\cdot b^n$

This law states that when an exponent is applied to parentheses containing a product of terms, the exponent is distributed to each factor inside the parentheses.

The second law we'll use is the power of a power law:

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

This law states that when an exponent is applied to a term that is already raised to a power (whether written with parentheses for clarity or not), the exponents are multiplied together.

Now, let’s return to the problem and first simplify the two terms with parentheses separately before handling the overall multiplication.

1. The first from left to right is:

$(xyz)^2=x^2\cdot y^2\cdot z^2$

When we expanded the parentheses using the first law, we applied the exponent to each factor in the product inside the parentheses.

2. The second from left to right is:

$(y^x)^y=y^{x y}$

When we expanded the parentheses using the second law, we obtained the following:

$y^3\cdot x^4\cdot(xyz)^2\cdot (y^x)^y=y^3\cdot x^4\cdot x^2\cdot y^2\cdot z^2\cdot y^{xy}$

When we used 1 and 2 mentioned above.

From here on we will no longer indicate the multiplication sign, but use the conventional writing form where placing terms next to each other means multiplication.

Now let's arrange the expression by bases using the multiplication commutative law:

$y^3x^4x^2y^2z^2y^{xy}=y^3y^2y^{xy}x^4x^2z^2$

Let's continue and simplify the expression by using the law of exponents for multiplication between terms with identical bases:

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

Note that this law is valid for any number of terms in multiplication and not just for two, for example for multiplication of three terms with identical bases 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 exponent law twice, we can also perform the same calculation for four terms in multiplication five and so on..

Let's apply this law to the problem:

$y^3y^2y^{xy}x^4x^2z^2=y^{3+2+xy}x^{4+2}z^2=y^{5+xy}x^6z^2$

When we used the above exponent law for multiplication between terms with identical bases only for terms with the same base.

We got the most simplified expression, so we're done.

Therefore the correct answer is C (when taking into account the commutative law of multiplication).

Important note:

It's worth understanding the reason for the power of a power law mentioned above (the second law), this law comes directly from the definition of exponents:

$(a^m)^n=a^m\cdot a^m\cdot\ldots\cdot a^m=a^{m+m+m+\cdots+m}=a^{m\cdot n}$

When in the first stage we applied the definition of exponents to the term in parentheses and multiplied it by itself n times, then we applied the above law of exponents for multiplication between terms with identical bases and interpreted the multiplication between terms as a sum in the exponent,

Then we used the simple multiplication definition stating that if we connect a number to itself n times we can simply write it as multiplication, meaning:

$m+m+\cdots+m=m\cdot n$

And therefore we get that:

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