Multiplication of Powers: Combination of different bases

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.

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

• Step 1: Identify terms with the same base.
• Step 2: Apply exponent rules to combine terms with the same base.
• Step 3: Rewrite the expression in its simplified form.

Now, let's work through each step:
Step 1: Identify terms with the same base:
The original expression $2^3 \times 3^8 \times 3^9 \times 2^6$ contains two bases: 2 and 3.

Step 2: Apply exponent rules:
For base 2: $2^3 \times 2^6 = 2^{3+6} = 2^9$.
For base 3: $3^8 \times 3^9 = 3^{8+9} = 3^{17}$.

Step 3: Rewrite the expression in simplified form:
The expression simplifies to $2^9 \times 3^{17}$.

Therefore, the solution to the problem is $2^9 \times 3^{17}$. Thus, choice 4 is correct.

To simplify the expression $6^2 \times 6^3 \times 3^2$, we will apply the rules for exponents:

• Step 1: Identify powers with the same base. Here, we notice that $6^2$ and $6^3$ are powers with the base 6.
• Step 2: Use the rule $a^m \times a^n = a^{m+n}$ to combine these powers: $6^2 \times 6^3 = 6^{2+3} = 6^5$.
• Step 3: The term $3^2$ remains unaffected by this operation because its base differs from that of 6. Therefore, it stays as $3^2$.

Therefore, the simplified form of $6^2 \times 6^3 \times 3^2$ is $6^5 \times 3^2$.

To simplify the given mathematical expression, we'll follow these steps:

• Step 1: Apply the Product of Powers Property to terms with the same base. For the expression $a^{-5} \times a^8$, we use the rule:
• $a^m \times a^n = a^{m+n}$.
• Step 2: Add the exponents: $-5 + 8 = 3$. Therefore, $a^{-5} \times a^8 = a^3$.
• Step 3: Since $x^3$ does not share the base $a$, it remains as is in the expression.

Therefore, the simplified form of the expression $a^{-5} \times a^8 \times x^3$ is:

$a^3 \times x^3$

To simplify the given expression $a^{-3} \times y^7 \times a^{-4} \times y^{-5}$, we will apply the exponent rules.

First, let's handle the terms involving the base $a$:

$a^{-3} \times a^{-4}$

According to the rule $x^m \times x^n = x^{m+n}$, we add the exponents:

$a^{-3} \times a^{-4} = a^{-3 + (-4)} = a^{-7}$

Next, consider the terms involving the base $y$:

$y^7 \times y^{-5}$

Using the same exponent rule:

$y^7 \times y^{-5} = y^{7 + (-5)} = y^2$

The entire expression now becomes:

$a^{-7} \times y^2$

Thus, the simplified form of the given expression is $a^{-7} \times y^2$.