Multiplication of Powers: Combination of different bases

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