Multiplication of Powers: Variable in the base of the power

Here, we will need to calculate a multiplication between terms with identical bases, therefore we will use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$Note that this property can only be used to calculate the multiplication between terms with identical bases,

We apply it to the problem:

$a^3\cdot a^4=a^{3+4}=a^7$Therefore, the correct answer is option b.

We will apply the law of exponents:

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

This means that when multiplying identical numbers raised to any power (meaning - identical bases raised to not necessarily identical powers), we can maintain the same base and simply add the exponents of the numbers,
Let's apply this law to the problem:

$a^4\cdot a^5=a^{4+5}=a^9$

Something important to remember is that this solution can also be explained verbally. Raising to a power effectively means multiplying the number (base) by itself as many times as the exponent indicates. Therefore multiplying $a$ by itself 4 times and multiplying the result by the result of multiplying $a$ by itself 5 times is like multiplying $a$ by itself 9 times, meaning multiplication between identical numbers (identical bases) raised to powers, not necessarily identical, can be calculated by keeping the same base (same number) and adding the exponents together.

To solve the given problem, we need to simplify the expression $a^2 \times a^3$ using the rules of exponents.

We use the rule for multiplying powers with the same base, which states:

• If you have $a^m \times a^n$, the result is $a^{m+n}$.

Let's apply this rule to the given expression:

$a^2 \times a^3 = a^{2+3}$

Simplifying the exponents, we get:

$a^{2+3} = a^5$

In the context of choosing from the given options, the answer corresponding to the application of the multiplication rule before final simplification is:

$a^{2+3}$

To solve the problem of simplifying $a^2 \times a^3 \times a^4$, we apply the exponent rule for multiplying powers with the same base.

This rule states that to multiply powers with the same base, we add their exponents:

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

Applying this rule to the problem at hand:

• The given expression is $a^2 \times a^3 \times a^4$

• We recognize that all parts have the same base, so we can add the exponents together: $2 + 3 + 4$.

• Therefore, we simplify the expression to $a^{2+3+4}$.

This matches with choice 4, $a^{2+3+4}$.

To reduce the equation $b^9 \times b^4 \times b^5$, follow these steps:

• Step 1: Identify the exponents present: $9$, $4$, and $5$.
• Step 2: Apply the exponent multiplication rule: Since the bases are the same, add the exponents: $9 + 4 + 5 = 18$.
• Step 3: Write the expression in its simplified form with the new exponent: $b^{18}$.

Therefore, the simplified form of the expression is $b^{18}$.

To solve this problem, we need to simplify the expression using the rules of exponents:

• Step 1: Recognize the base $b$ is the same for both terms in the multiplication.

• Step 2: Apply the exponent multiplication rule: add the exponents of like bases. Thus, $b^4 \times b^5 = b^{4+5}$.

The correct answer to the problem is$b^{4+5}$.

To solve the problem of simplifying $t^7 \times t^2$, we follow these steps:

• Step 1: Identify the base and the exponents. Here, the base is $t$, the first exponent is 7, and the second exponent is 2.
• Step 2: Apply the exponent rule for multiplying powers with the same base, which states $a^m \times a^n = a^{m+n}$.
• Step 3: Add the exponents: $7 + 2 = 9$.
• Step 4: Write the simplified expression: $t^9$.

Therefore, after applying the exponent rule, the simplified form of the expression is $t^9$.

The correct choice among the given options is not specifically listed, but the simplification corresponds to $t^{7+2}$ before explicitly adding to get $t^9$.

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

• Step 1: Identify the base and the exponents from the given expression.
• Step 2: Apply the rule for multiplying powers with the same base.
• Step 3: Perform the addition to simplify the exponents.

Now, let's work through each step:
Step 1: The expression given is $t^6 \times t^7$. The base here is $t$, and the exponents are 6 and 7.

Step 2: According to the rule for multiplying exponents with the same base, we add the exponents. Therefore, the expression becomes:
$t^6 \times t^7 = t^{6+7}$.

Step 3: Simplify the expression by adding the exponents:
$6 + 7 = 13$.

Therefore, the simplified expression is $t^{13}$.

The correct choice given is: $t^{13}$, which is choice 2.

To solve this problem, follow these steps:

• Step 1: Identify the exponents in the expression $x^8 \times x^7 \times x^{10}$. They are 8, 7, and 10.
• Step 2: Apply the multiplication rule for exponents, which states $a^m \times a^n = a^{m+n}$.
  Here, it becomes: $x^8 \times x^7 \times x^{10} = x^{8+7+10}$.
• Step 3: Simplify the expression by adding the exponents together:

After performing the addition, $8 + 7 + 10 = 25$.

Thus, the reduced form of the equation is $x^{25}$.

Therefore, the final answer is $x^{25}$.

Here we will have to to multiply terms with identical bases, therefore we use the appropriate power property:

$b^m\cdot b^n=b^{m+n}$Note that this property can only be used to calculate the multiplication between terms with identical bases,

From now on we no longer write the multiplication sign, but use the accepted form of writing in which placing terms next to each other means multiplication.

We apply it in the problem:

$x^2x^5=x^{2+5}=x^7$Therefore, the correct answer is D.

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

• Step 1: Identify the exponents in the expression $y^2 \cdot y^3 \cdot y^6$.

• Step 2: Apply the exponent rule by adding the exponents together.

• Step 3: Simplify the combined exponents to find the final expression.

Now, let's work through each step:

Step 1: The exponents in the expression are 2, 3, and 6.

Step 2: According to the multiplication rule for powers with the same base, we have $y^2 \cdot y^3 \cdot y^6 = y^{2+3+6}$.

Step 3: Calculate the sum of the exponents: $2 + 3 + 6 = 11$.

Therefore, the simplified expression is $y^{11}$.

Given the choices, the correct answers, by these computations, correspond to:

• $y^{11}$

• $y^{2+3+6}$

Hence, the correct answer to the problem is B+C are correct.

Note that we need to calculate multiplication between terms with identical bases, so we'll use the appropriate exponent law:

$b^m\cdot b^n=b^{m+n}$Note that we can only use this law to calculate multiplication performed between terms with identical bases,

Here in the problem there is also a term with a negative exponent, but this does not pose an issue regarding the use of the aforementioned exponent law. In fact, this exponent law is valid in all cases for numerical terms with different exponents, including negative exponents, rational number exponents, and even irrational number exponents, etc.,

Let's apply it to the problem:

$y^{-2}\cdot y^7=y^{-2+7}=y^5$Therefore the correct answer is A.

To solve the problem of simplifying the expression $y^9 \times y^2 \times y^3$, we will follow these steps:

First, recognize that the expression entails powers of the same base $y$, and we can use the rule for multiplying powers with the same base. This rule states that when multiplying like bases, we add the exponents. Mathematically, this can be expressed as:

• Step 1: Identify the base $y$ and the exponents $9$, $2$, and $3$.
• Step 2: Apply the rule for multiplication of powers $y^9 \times y^2 \times y^3 = y^{9+2+3}$.
• Step 3: Calculate the sum of the exponents: $9 + 2 + 3 = 14$.
• Step 4: Simplify the expression to yield the solution, $y^{14}$.

In reviewing the answer choices:

• Choice 1: $y^{9+2+3}$ represents the fully simplified expression, which agrees with our solution.
• Choice 2: $y^{9+2} \times y^3$ and choice 3: $y^9 \times y^{2+3}$ still maintain some intermediate steps of simplification. Yet, both can eventually be simplified further to $y^{14}$.

Therefore, all expressions represent correct approaches or intermediates toward achieving the correct final form. Thus, All answers are correct.

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$It is important to note that this property is only valid for terms with identical bases,

We return to the problem

We notice that in the problem there are two types of terms with different bases. First, for the sake of order, we will use the substitution property of multiplication to rearrange the expression so that the two terms with the same base are grouped together. Then, we will proceed to work:

$a\cdot b\operatorname{\cdot}a\operatorname{\cdot}b\operatorname{\cdot}a^2=a\cdot a\cdot a^2\cdot b\cdot b$Next, we apply the power property for each type of term separately,

$a\cdot a\cdot a^2\cdot b\cdot b=a^{1+1+2}\cdot b^{1+1}=a^4\cdot b^2$

$$We apply the power property separately - for the terms whose bases are$a$and then for the terms whose bases are$b$and we add the exponents and simplify the terms.

$$Therefore, the correct answer is option c.

Note:

We use the fact that:

$a=a^1$and the same for $b$.

To solve this problem, we begin by rewriting the expression that incorporates exponent rules. The expression given is $a^{3+5}$. According to the rule of exponents, when you have a base raised to a power that is a sum, $a^{m+n} = a^m \times a^n$.

Let's apply this rule:

• Write the exponent as a sum: $3 + 5$.
• Apply the exponent rule: $a^{3+5}$ becomes $a^3 \times a^5$.

Thus, the expanded form of $a^{3+5}$ using the rule of exponents is $a^3 \times a^5$.

Finally, comparing with the provided options, choice 1 ( $a^3 \times a^5$) is the correct one, as it correctly uses the exponent rule.

Therefore, the solution to the problem is $a^3\times a^5$.

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:

$a^5\times b^9$

Let's start with multiplying the fractions, remembering that the multiplication of fractions is performed by multiplying the numerator by numerator and the denominator by the denominator:

$\frac{b^{22}}{b^{20}}\cdot\frac{b^{30}}{b^{20}}=\frac{b^{22}\cdot b^{30}}{b^{20}\cdot b^{20}}$

In both the numerator and denominator, multiplication occurs between terms with identical bases, so we'll apply the power law for multiplying terms with identical bases:

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

This law can only be used when multiplication is performed between terms with identical bases.

From here on, we will no longer indicate the multiplication sign, instead we will place terms next to each other.
Let's return to the problem and apply the above power law separately to the fraction's numerator and denominator:

$\frac{b^{22}b^{30}}{b^{20}b^{20}}=\frac{b^{22+30}}{b^{20+20}}=\frac{b^{52}}{b^{40}}$

In the final step we calculated the sum of the exponents in the numerator and denominator.

Note that division is required between two terms with identical bases, hence we'll apply the power law for division between terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$

This law can only be used when division is performed between terms with identical bases.

Let's return to the problem and apply the above power law:

$\frac{b^{52}}{b^{40}}=b^{52-40}=b^{12}$

In the final step we calculated the subtraction between the exponents.

This is the most simplified form of the expression:

Therefore, the correct answer is C.

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

• Step 1: Identify the power that needs expansion: $b^7$.
• Step 2: Use the rule for the multiplication of powers to express this as a product of smaller powers of $b$.
• Step 3: Decompose 7 into a sum of smaller numbers and express the power as a product of smaller terms.

Now, let's apply these steps:

Step 1: We have $b^7$ and need to write it as a product of powers.

Step 2: Recall the rule for multiplication of powers, $b^a \times b^b = b^{a+b}$. We will express 7 as the sum of smaller numbers.

Step 3: Choose smaller exponents that add up to 7. Here, we can use $1 + 2 + 4 = 7$. Therefore, $b^7 = b^1 \times b^2 \times b^4$.

Therefore, the expanded form of the expression is $b^1 \times b^2 \times b^4$.

Apply the power rule for multiplying terms with identical bases:

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

Note that this rule is valid only for terms with identical bases,

Here in the problem there are also terms with negative exponents, but this doesn't pose an issue regarding the use of the aforementioned power rule. In fact, this power rule is valid in all cases for numerical terms with different powers, including negative powers, rational number powers, and even irrational number powers, etc.

Let's return to the problem,

Note that there are two types of terms in the problem that differ from each other with different bases. First, we'll apply the commutative law of multiplication to arrange the expression so that all terms with the same base are adjacent, let's get to work:

$c^{-1}\cdot d^6\cdot d^{-2}\cdot c^3\cdot c^2=c^{-1}\cdot c^3\cdot c^2\cdot d^6\cdot d^{-2}$

Then we'll proceed to apply the aforementioned power rule separately to each different type of term,

$c^{-1}\cdot c^3\cdot c^2\cdot d^6\cdot d^{-2}=c^{-1+3+2}\cdot d^{6+(-2)}=c^{-1+3+2}\cdot d^{6-2}=c^4\cdot d^4$

$$To summarise we applied the above rule separately - for terms with base $c$ and for terms with base $d$ and then combined the powers in the exponent when we grouped all terms with the same base together.

Therefore, the correct answer is B.

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$It should be noted that this property is only valid for terms with identical bases,

We return to the problem

We notice that in the problem there are two types of terms with different bases. First, for the sake of order, we will use the substitution property of multiplication to rearrange the expression so that the two terms with the same base are grouped together. Then, we will proceed to work:

$E^6\cdot F^{-4}\cdot E^0\cdot F^7\cdot E=E^6\cdot E^0\cdot E\cdot F^{-4}\cdot F^7$Next, we apply the power property for each type of term separately,

$E^6\cdot E^0\cdot E\cdot F^{-4}\cdot F^7=E^{6+0+1}\cdot F^{-4+7}=E^7\cdot F^3$

$$We apply the power property separately - for the terms whose bases are$E$and for the terms whose bases are$F$and we add the exponents and simplify the terms with the same base.

The correct answer is then option d.

Note:

We use the fact that:

$E=E^1$.