Multiplication of Powers Practice Problems & Solutions

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.

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 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 the problem of simplifying $15^2 \times 15^4$, we will use the rule for multiplying exponents with the same base.

According to the multiplication of powers rule: If $a$ is a real number and $m$ and $n$ are integers, then:

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

Applying this rule to our problem, where the base $a$ is 15, and the exponents $m$ and $n$ are 2 and 4 respectively:

• Step 1: Identify the base and exponents: $15^2$ and $15^4$ have the same base.
• Step 2: Add the exponents: $2 + 4 = 6$.
• Step 3: Simplify the expression using the rule: $15^2 \times 15^4 = 15^{2+4} = 15^6$.

Therefore, the simplified expression is $15^6$.

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: $2^{2+3}$.