Multiplication of Powers - Examples, Exercises and Solutions

To solve the exercise we use the property of multiplication of powers with the same bases:

$a^n * a^m = a^{n+m}$

With the help of this property, we can add the exponents.

$4^2\times4^4=4^{4+2}=4^6$

To simplify the expression $5^8 \times 5^3$, we will use the exponent rule which states that when multiplying powers with the same base, we add the exponents:

Step-by-step:

• Identify the base and exponents: Here, the base is $5$, and the exponents are $8$ and $3$.

• Apply the multiplication of exponents rule: $5^8 \times 5^3 = 5^{8+3}$.

Therefore, the correct answer is $5^{8+3}$.

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

• Step 1: Identify the given expression and its components.

• Step 2: Apply the exponent multiplication formula.

• Step 3: Simplify the result.

Now, let's work through each step:
Step 1: The given expression is $3^4 \times 3^5$. We recognize that the base is 3 and the exponents are 4 and 5.
Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$. Using this formula, we add the exponents: $4 + 5$.
Step 3: Simplify the expression: $3^{4+5} = 3^9$.

Therefore, the simplified form of the expression is $3^{4+5}$.

To simplify the expression $2^2 \times 2^3$, we apply the rule for multiplying powers with the same base. According to this rule, when multiplying two exponential expressions that have the same base, we keep the base and add the exponents.

• Step 1: Identify the base: In this problem, the base for both terms is 2.
• Step 2: Apply the exponent multiplication rule: $2^2 \times 2^3 = 2^{2+3}$.
• Step 3: Simplify by adding the exponents: $2^{2+3} = 2^5$.

Thus, the simplified form of the expression $2^2 \times 2^3$ is $2^{5}$.

The correct choice from the provided options is: $2^{2+3}$.

To solve this problem, let's apply the multiplication of powers rule:

• Step 1: Identify expression as $9 \times 9^9$.
• Step 2: Note that $9$ can be expressed as $9^1$.
• Step 3: Apply the exponent rule: $a^m \times a^n = a^{m+n}$.

Now, we'll work through the calculation step-by-step:

Step 1: Rewrite $9$ as $9^1$. Thus, our expression becomes $9^1 \times 9^9$.

Step 2: Use the exponent rule to combine: $9^1 \times 9^9 = 9^{1+9}$.

Step 3: Simplify the exponent by adding: $9^{1+9} = 9^{10}$.

Therefore, the simplified form of the expression is $9^{10}$.

In terms of the answer choices, the correct answer is

$9^{1+9}$

To simplify the expression $8^3 \times 8$, we begin by identifying the implicit exponent for the standalone 8. Since there is no written exponent next to the second 8, we can assume it has an exponent of 1.

Thus, the expression can be written as:\br $8^3 \times 8^1$.

Using the rule for multiplying powers with the same base, $a^m \times a^n = a^{m+n}$, we add the exponents:

• Here, the base $a$ is 8.
• The exponents are 3 and 1.

Therefore, $8^3 \times 8^1 = 8^{3+1} = 8^4$.

Thus, the simplified expression is $\mathbf{8^4}$.

Consequently, the correct choice is $8^{3+1}$.

To simplify the expression given, $6^2 \times 6^8$, we will use the property of exponents which states that the product of two powers with the same base is the base raised to the sum of the exponents.

Let's apply the rule:

• The base in both powers is $6$.

• The exponents are $2$ and $8$.

• According to the rule $a^m \times a^n = a^{m+n}$, we add the exponents; therefore, $6^2 \times 6^8 = 6^{2+8}$.

• Simplifying further, this becomes $6^{10}$.

Therefore, the simplified expression is $6^{10}$.

The solution to the given problem is $6^{2+8}$.

To simplify the equation $12 \times 12^2$, follow these steps:

• Step 1: Recognize that 12 can be expressed as a power. Since $12 = 12^1$, rewrite the equation as $12^1 \times 12^2$.
• Step 2: Apply the rule for multiplying powers with the same base, which states that $a^m \times a^n = a^{m+n}$. In this case, this becomes $12^{1+2}$.
• Step 3: Simplify the expression by adding the exponents: $1 + 2 = 3$.

Thus, the simplified form of the expression is $12^3$.

Therefore, the correct answer choice is $12^{1+2}$, which corresponds to choice 2.

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

• Step 1: Identify the given expression.

• Step 2: Recognize and apply the exponent multiplication rule.

• Step 3: Simplify the expression by adding the exponents.

Now, let's work through each step:
Step 1: The expression given is $7^6 \times 7^6$.
Step 2: Since the bases are the same, apply the exponent rule: $a^m \times a^n = a^{m+n}$.
Step 3: By adding the exponents, we have $6 + 6 = 12$.

Therefore, the simplified expression is $7^{6+6}$ or $7^{12}$.

This corresponds to choice 2.

Thus, the solution to the problem is $7^{6+6}$.

To solve the problem of simplifying the equation $11^{10} \times 11^{11}$, follow these steps:

• Step 1: Identify that the bases are the same (11).

• Step 2: Apply the multiplication of powers rule, which states that when multiplying like bases, you add the exponents.

• Step 3: Add the exponents: $10 + 11$.

• Step 4: Perform the addition: $10 + 11 = 21$.

• Step 5: Write the expression with the new exponent: $11^{10+11}= 11^{21}$.

Therefore, the simplified expression is $11^{21}$. This corresponds to options 1 and 2 being correct as they represent the same expression when evaluating the sum, which is also represented by choice 4 as "a'+b' are correct".

To solve this problem, we'll simplify the expression $9^2 \times 9^9$ using the multiplication of powers rule:

• Step 1: Identify the base and the exponents. The base here is 9, and the exponents are 2 and 9.
• Step 2: Apply the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$.
• Step 3: Add the exponents: $9^2 \times 9^9 = 9^{2+9} = 9^{11}$.

The expression simplifies to $9^{11}$.

Therefore, the simplified expression is $9^{11}$, which matches choice 1.

To solve this problem, we'll use the properties of exponents to simplify the expression:

• Step 1: Identify the bases and exponents in the expression $8^3 \times 8^6$.
• Step 2: Apply the product of powers property, which states $a^m \times a^n = a^{m+n}$ when the bases are the same.
• Step 3: Add the exponents: $3 + 6 = 9$.

Now, let's work through these steps:

Step 1: Both terms, $8^3$ and $8^6$, have the same base, 8.

Step 2: According to the product of powers property, we add the exponents: $8^{3+6}$.

Step 3: Simplifying the exponents gives us $8^9$.

Therefore, the simplified expression is $8^9$.

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

• Step 1: Identify the given expression
• Step 2: Apply the exponent rule for multiplication of powers with the same base
• Step 3: Simplify the expression by adding the exponents

Now, let's work through each step:
Step 1: The given expression is $6^5 \times 6^7$. Here, the base is 6, and the exponents are 5 and 7.
Step 2: We apply the exponent rule, which states that when multiplying two powers with the same base, we add the exponents. Therefore, we have:

$6^5 \times 6^7 = 6^{5+7}$

Step 3: Add the exponents: $5 + 7 = 12$. Thus, the expression simplifies to:

$6^{12}$

Therefore, the solution to the problem is $6^{12}$.

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

• Step 1: Identify the base and exponents
• Step 2: Apply the exponent multiplication rule
• Step 3: Perform the calculations

Let's work through each step:
Step 1: We have $3^2$ and $3^3$. Both have the same base, which is 3.
Step 2: According to the exponent multiplication rule $a^m \times a^n = a^{m+n}$, we add the exponents:
$2 + 3 = 5$.
Step 3: Rewrite the expression as a single power:
$3^2 \times 3^3 = 3^{2+3} = 3^5$.

Therefore, the simplified expression is $\boldsymbol{3^5}$, which corresponds to choice 2.

To solve this problem, let’s follow the outlined steps:

• Step 1: Calculate $10 \times 10$.
• Step 2: Express the calculation using exponent rules.
• Step 3: Verify the possible expressions match the given choices.

Now, let's work through each step:
Step 1: The direct multiplication of 10 by 10 yields $100$ because $10 \times 10 = 100$.
Step 2: We can express this calculation using the rules of exponents. Since both numbers are 10 and multiplied together: $10^1 \times 10^1 = 10^{1+1} = 10^2$.
Step 3: We consider the following expressions given in the multiple-choice answers:
- Choice 1: $10^{1+1}$ equals $10^2$.
- Choice 2: $10^2$ equals 100.
- Choice 3: 100 is the result of the direct multiplication.

Each choice is consistent with the others through these steps. Thus, all the provided expressions—$10^{1+1}$, $10^2$, and 100—accurately represent the resolved equation $10 \times 10$.

Therefore, the solution to the problem is All answers are correct.