Multiplication of Powers - Examples, Exercises and Solutions

To simplify the expression $4^5 \times 4^5$, we will use the rule of exponents that states when multiplying two powers with the same base, you can add the exponents. This rule can be expressed as:

• $a^m \times a^n = a^{m+n}$

In this equation, both terms $4^5$ have the same base $4$.

According to the multiplication of powers rule:

• $4^5 \times 4^5 = 4^{5+5}$

Now, simply add the exponents:

$4^{5+5} = 4^{10}$

The simplified form of $4^5 \times 4^5$ is therefore $4^{10}$.

To solve the expression $7^9 \times 7^1$, we need to apply the rules of exponents, specifically the multiplication of powers rule. According to this rule, when we multiply two powers with the same base, we keep the base and add the exponents together.

• The expression can be written as $a^m \times a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
• In this case, our base $a$ is 7, and our exponents are 9 and 1.
• Applying the formula, we have: $7^9 \times 7^1 = 7^{9+1}$.
• Simplifying the exponent: $9 + 1 = 10$.

Thus, the expression simplifies to: $7^{10}$.

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.

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

All bases are equal and therefore the exponents can be added together.

$8^2\cdot8^3\cdot8^5=8^{10}$

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

$a^m\cdot a^n=a^{m+n}$Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

Keep in mind that all the terms in the multiplication have the same base, so we will use the previous property:

$2^{10}\cdot2^7\cdot2^6=2^{10+7+6}=2^{23}$Therefore, the correct answer is option c.

According to the property of powers, when there are two powers with the same base multiplied together, the exponents should be added.

According to the formula:$a^n\times a^m=a^{n+m}$

It is important to remember that a number without a power is equivalent to a number raised to 1, not to 0.

Therefore, if we add the exponents:

$7^{9+1}=7^{10}$

To expand the equation $3^{12+10+5}$, we will apply the rule of exponents that states: when you multiply powers with the same base, you can add the exponents. However, in this case, we are starting with a single term and want to represent it as a product of terms with the base being raised to each of the individual exponents given in the sum. Here’s a step-by-step explanation:

1. Start with the expression: $3^{12+10+5}$.

2. Recognize that the exponents are added together. According to the property of exponents (Multiplication of Powers), we can express a single power with summed exponents as a product of powers:

3. Break down the exponents: $3^{12+10+5} = 3^{12} \times 3^{10} \times 3^5$.

4. As seen from the explanation: $3^{12+10+5}$ is expanded to the product $3^{12} \times 3^{10} \times 3^5$ by expressing each part of the sum as an exponent with the base 3.

The final expanded form is therefore: $3^{12} \times 3^{10} \times 3^5$.

To solve the expression $6^3 \times 6^{-4} \times 6^7$, we need to apply the rules of exponents, specifically the multiplication of powers. When we multiply powers with the same base, we add their exponents.

First, let's identify the base and the exponents in the expression:

• The base is 6.

• The exponents are 3, -4, and 7.

Using the exponent multiplication rule, we sum the exponents:

$6^3 \times 6^{-4} \times 6^7= 6^{3 + (-4) + 7}$

So, the solution is:

$6^{3-4+7}$

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.

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.

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.

We will use the law of exponents:

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

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

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

Let's note something important, that this solution can also be explained verbally, since raising to a power means multiplying the number (base) by itself as many times as the exponent indicates, and 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.

We begin by using the rule for multiplying exponents. (the multiplication between terms with identical bases):

$a^m\cdot a^n=a^{m+n}$We then apply it to the problem:

$7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1}$When in a first stage we begin by applying the aforementioned rule and then continue on to simplify the expression in the exponent,

Next, we use the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$We apply it to the expression obtained in the previous step:

$7^{-1}=\frac{1}{7^1}=\frac{1}{7}$We then summarise the solution to the problem: $7^5\cdot7^{-6}=7^{-1}=\frac{1}{7}$Therefore, the correct answer is option B.

To solve the expression $4^3 \times 4^{-5}$, we need to apply the multiplication of powers rule. This rule states that when you multiply two powers with the same base, you can add their exponents. Mathematically, this is expressed as:

• $a^m \times a^n = a^{m+n}$

In our case, the base $a$ is 4, and the exponents $m$ and $n$ are 3 and -5, respectively.

Applying the rule:

$4^3 \times 4^{-5} = 4^{3 + (-5)}$

Simplifying the exponent:

$3 + (-5) = -2$

So, the expression simplifies to:

$4^{-2}$

This is the reduced form of the given equation.

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of $3$

$3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}$