Multiplication of Powers: Applying the formula

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:

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

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.

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.

To solve this exercise, first we note that 25 is the result of a power and we reduce it to a common base of 5.

$\sqrt{25}=5$$25=5^2$Now, we go back to the initial exercise and solve by adding the powers according to the formula:

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

$5^4\times25=5^4\times5^2=5^{4+2}=5^6$

We use the law of exponents to multiply terms with identical bases:

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

$7^x\cdot7^{-x}=7^{x+(-x)}=7^{x-x}=7^0$In the first stage we apply the above power rule and in the following stages we simplify the expression obtained in the exponent,

Subsequently, we use the zero power rule:

$X^0=1$We obtain:

$7^0=1$Lastly we summarize the solution to the problem.

$7^x\cdot7^{-x}=7^{x-x}=7^0 =1$Therefore, the correct answer is option B.

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 of the multiplication have the same base, so we will use the previous property:

$5^{-3}\cdot5^0\cdot5^2\cdot5^5=5^{-3+0+2+5}=5^4$Therefore, the correct answer is option c.

Note:

Keep in mind that $5^0=1$

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:

First keep in mind that:

$10=10^1$Keep in mind that all the terms of the multiplication have the same base, so we will use the previous property:

$10^1\cdot10^2\cdot10^{-4}\cdot10^{10}=10^{1+2-4+10}=10^9$

$$Therefore, the correct answer is option c.