Multiplication of Powers: Presenting powers with negative exponents as fractions

To solve the problem $300^{-4} \cdot \left(\frac{1}{300}\right)^{-4}$, let's follow these steps:

• Step 1: Simplify the reciprocal power expression.
  The expression $\left(\frac{1}{300}\right)^{-4}$ can be simplified using the rule for reciprocals, which states $\left(\frac{1}{a}\right)^{-n} = a^n$. Thus, $\left(\frac{1}{300}\right)^{-4} = 300^4$.
• Step 2: Combine the powers.
  Now the expression becomes $300^{-4} \cdot 300^4$. Using the rule $a^m \cdot a^n = a^{m+n}$, we have $300^{-4+4} = 300^0$.
• Step 3: Calculate the final result.
  By the identity $a^0 = 1$, any non-zero number raised to the power of zero equals 1. Therefore, $300^0 = 1$.

Thus, the solution to the problem is $\boxed{1}$.

To solve the expression $11^{-2} \times 11^{-5} \times 11^{-4}$, we apply the rules for multiplying numbers with the same base:

• Step 1: Use the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$.

• Step 2: Add the exponents: $-2 + -5 + -4$.

• Step 3: Perform the calculation: $-2 - 5 - 4 = -11$.

• Step 4: Write the expression with the combined exponent: $11^{-11}$.

• Step 5: Express $11^{-11}$ as a positive power using the property of negative exponents: $a^{-n} = \frac{1}{a^n}$.

Therefore, $11^{-11} = \frac{1}{11^{11}}$.

The final answer is $\frac{1}{11^{11}}$.

Let's simplify the expression $5^{-8} \times 5^6$ using the rules of exponents.

• Step 1: Apply the rule for multiplying powers with the same base. According to this rule, when multiplying like bases, we add the exponents: $5^{-8} \times 5^6 = 5^{-8 + 6}$
• Step 2: Calculate the sum of the exponents: $-8 + 6 = -2$.
• Step 3: Write the simplified expression: $5^{-2}$.
• Step 4: Relate the expression to the given choices:

The simplified expression $5^{-2}$ corresponds to choice 1. Additionally, rewriting a negative exponent using the fraction format gives: $5^{-2} = \frac{1}{5^2}$, which matches choice 2.

Thus, both choices 'a: $5^{-2}$' and 'b: $\frac{1}{5^2}$' are correct.

Therefore, according to the given answer choice, a'+b' are correct.

To simplify the expression $4^{-6} \times 4$, follow these steps:

• Step 1: Apply the rule for multiplying powers with the same base, which is $a^m \times a^n = a^{m+n}$.

• Step 2: Identify the exponents for the terms. Here, we have $4^{-6}$ and $4^1$, implying $m = -6$ and $n = 1$.

• Step 3: Add the exponents: $(-6) + 1 = -5$. Thus, we have $4^{-6} \times 4^1 = 4^{-5}$.

• Step 4: Recognize that a negative exponent indicates a reciprocal. Therefore, $4^{-5} = \frac{1}{4^5}$.

Therefore, the solution to the expression $4^{-6} \times 4$ is $\frac{1}{4^5}$.

To solve this problem, we need to simplify the expression $8^{-10} \times 8^{-5} \times 8^9$ using exponent rules.

• Step 1: Apply the multiplication rule for exponents. This rule states that when multiplying expressions with the same base, you add their exponents. Thus, we calculate:
  $8^{-10} \times 8^{-5} \times 8^9 = 8^{-10 + (-5) + 9}$.
• Step 2: Simplify the exponents:
  $-10 + (-5) + 9 = -10 - 5 + 9 = -6$.
• Step 3: The expression simplifies to:
  $8^{-6}$.
• Step 4: Convert the negative exponent into a positive one by using the rule for negative exponents, where $a^{-n} = \frac{1}{a^n}$:
  $8^{-6} = \frac{1}{8^6}$.

Therefore, the simplified expression is $\frac{1}{8^6}$.

The corresponding expression is:

To solve the problem $9^{-1} \times 9^{-2} \times 9^{-3}$, we follow these steps:

• Step 1: Use the rule for multiplying exponential terms with the same base. The formula is $a^m \times a^n = a^{m+n}$.
• Step 2: Apply the formula to the given expression: $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-1-2-3}$.
• Step 3: Simplify the exponent: $-1 - 2 - 3 = -6$. Therefore, $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-6}$.

We can express $9^{-6}$ as a positive power by recalling that negative exponents indicate reciprocals:

$9^{-6} = \frac{1}{9^6}$.

Thus, both $9^{-6}$ and $\frac{1}{9^6}$ are valid expressions for the simplified form. Additionally, the expression $9^{-1-2-3}$ highlights the step where we combined the exponents, and it is equivalent to the final result. Therefore, all given answers correctly represent the simplified expression.

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

$$We will now apply the law of negative exponents, but in reverse:

$\frac{1}{a^x} =a^{-x}$

We'll apply this law to the problem for the third term in the product:

$m^{-n}\cdot n^{-m}\cdot\frac{1}{m}=m^{-n}\cdot n^{-m}\cdot m^{-1}=m^{-n}\cdot m^{-1}\cdot n^{-m}$

When in the first stage we applied the above law of exponents for the third term in the product, and in the next stage we rearranged the resulting expression using the distributive property of multiplication so that terms with identical bases are adjacent to each other,

Next, we'll recall the law of exponents for multiplying terms with identical bases:

$a^x\cdot a^y=a^{x+y}$

And we'll apply this law of exponents to the expression that we obtained in the last stage:

$m^{-n}\cdot m^{-1}\cdot n^{-m}=m^{-n+(-1)}\cdot n^{-m}=m^{-n-1}\cdot n^{-m}$

When in the first stage we applied the above law of exponents for terms with identical bases, and in the next stage we simplified the expression with the exponent of the first term in the product in the expression we obtained the following,

Let's summarize the solution so far as shown below:

$m^{-n}\cdot n^{-m}\cdot\frac{1}{m}=m^{-n}\cdot n^{-m}\cdot m^{-1}=m^{-n-1}\cdot n^{-m}$

Now let's note that there is no such answer among the given options, and an additional check of what we've done so far will reveal that there is no calculation error,

Therefore, we can conclude that additional mathematical manipulation is needed to determine which is the correct answer among the given options,

Let's note that options B and D have expressions similar to the expression we got in the last stage, while the other two options can be directly eliminated since they are clearly different from the expression we got,

Furthermore, let's note that in addition, the second term in the product in the expression we got, which is the term-$n^{-m}$, is in the numerator (note at the end of the solution on this topic), while in option B it's in the denominator, so we'll eliminate this option,

Thus - we're left with only one option - which is answer D, however we want to verify (and must verify!) that this is indeed the correct answer:

We'll do this using the law of exponents for negative exponents that we mentioned earlier, but in the forward direction:

$a^{-x} = \frac{1}{a^x}$

And we'll deal separately with the first term in the product in the expression we got in the last stage of solving the problem, which is the term:

$m^{-n-1}$

Let's note that we can represent the expression in the exponent as follows:

$-n-1=-(n+1)$

Where we used factoring out and took out negative one from the parentheses,

Next, we'll use the above law of exponents and the last understanding to represent the above expression (which we're currently dealing with, separately) as a term in the denominator of a fraction:

$m^{-n-1}=m^{-\underline {\bm{(n+1)}}}=\frac{1}{m^{\underline {\bm{n+1}}}}$

When in the first stage, in order to use the above law of exponents - we represented the term in question as having a negative exponent, while using the fact that:

$-n-1=-(n+1)$,

Next, we applied the above law of exponents carefully, since the number that- x represents in our use of the above law of exponents here is:

$n+1$(underlined in the expression above)

Let's return then to the expression we got in the last stage of solving the given problem, and apply for the first term in the product the mathematical manipulation we just performed:

$m^{-n-1}\cdot n^{-m}=m^{-(n+1)}\cdot n^{-m}=\frac{1}{m^{n+1}}\cdot n^{-m}$

Now let's simplify the expression that we obtained and perform the multiplication in the fraction while remembering that multiplication in a fraction means multiplying the numerators:

$\frac{1}{m^{n+1}}\cdot n^{-m}=\frac{1\cdot n^{-m}}{m^{n+1}}=\frac{n^{-m}}{m^{n+1}}$

Let's summarize then the solution stages so far as follows:

$m^{-n}\cdot n^{-m}\cdot\frac{1}{m}=m^{-n-1}\cdot n^{-m} =\frac{n^{-m}}{m^{n+1}}$

Therefore, the correct answer is indeed answer D.

Note:

When it's written "the number in the numerator" even though there isn't actually a fraction in the expression, this is because we can always refer to any number as being in the numerator of a fraction if we remember that any number divided by 1 equals itself, meaning, we can always write a number as a fraction like this:

$X=\frac{X}{1}$and therefore we can actually refer to $X$as a number in the numerator of a fraction.

First let's note that the number 9 is a power of the number 3:

$9=3^2$

Therefore we can immediately move to a unified base in the problem, in addition we'll recall the law of powers for negative exponents but in the opposite direction:

$\frac{1}{a^n} =a^{-n}$

Let's apply this to the problem:

$9^4\cdot3^{-8}\cdot\frac{1}{3}=(3^2)^4\cdot3^{-8}\cdot3^{-1}$

In the first term of the multiplication we replaced the number 9 with a power of 3, according to the relationship mentioned earlier, and simultaneously the third term in the multiplication we expressed as a term with a negative exponent according to the aforementioned law of exponents.

Now let's recall two additional laws of exponents:

a. The law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

b. The law of exponents for multiplication between terms with equal bases:

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

Let's apply these two laws to the expression we got in the last stage:

$(3^2)^4\cdot3^{-8}\cdot3^{-1}=3^{2\cdot4}\cdot3^{-8}\cdot3^{-1}=3^8\cdot3^{-8}\cdot3^{-1}=3^{8+(-8)+(-1)}=3^{8-8-1}=3^{-1}$

In the first stage we applied the law of exponents for power of a power mentioned in a', in the next stage we applied the law of exponents for multiplication of terms with identical bases mentioned in b', then we simplified the resulting expression.

Let's summarize the solution steps:

$9^4\cdot3^{-8}\cdot\frac{1}{3}=(3^2)^4\cdot3^{-8}\cdot3^{-1} =3^{8+(-8)+(-1)}=3^{8-8-1}=3^{-1}$

Therefore the correct answer is answer b'.

We'll use the law of exponents for negative exponents:

$a^{-n}=\frac{1}{a^n}$

Let's apply this law to the problem:

$5^4-(\frac{1}{5})^{-3}\cdot5^{-2}= 5^4-(5^{-1})^{-3}\cdot5^{-2}$

We apply the above law of exponents to the second term from the left.

Next, we'll recall the law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply this law to the expression that we obtained in the last step:

$5^4-(5^{-1})^{-3}\cdot5^{-2} = 5^4-5^{(-1)\cdot (-3)}\cdot5^{-2} = 5^4-5^{3}\cdot5^{-2}$

We apply the above law of exponents to the second term from the left and then simplify the resulting expression,

Let's continue and recall the law of exponents for multiplication of terms with the same base:

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

Let's apply this law to the expression that we obtained in the last step:

$5^4-5^{3}\cdot5^{-2} =5^4-5^{3+(-2)}=5^4-5^{3-2}=5^4-5^{1} =5^4-5$

We apply the above law of exponents to the second term from the left and then simplify the resulting expression,

From here, notice that we can factor the expression by taking out the common factor 5 from the parentheses:

$5^4-5 =5(5^3-1)$

Here we also used the law of exponents for multiplication of terms with the same base mentioned earlier, in the opposite direction:

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

Notice that:

$5^4=5\cdot 5^3$

Let's summarize the solution so far:

$5^4-(\frac{1}{5})^{-3}\cdot5^{-2}= 5^4-(5^{-1})^{-3}\cdot5^{-2} = 5^4-5^{3}\cdot5^{-2}=5(5^3-1)$

Therefore the correct answer is answer C.

Apply the laws of exponents for negative exponents, in the opposite direction:

$\frac{1}{a^n} = a^{-n}$

for the middle term in the multiplication in the problem:

$9^{300}\cdot\frac{1}{9^{-252}}\cdot9^{-549}=9^{300}\cdot9^{-(-252)}\cdot9^{-549}=9^{300}\cdot9^{252}\cdot9^{-549}$

In the first stage we'll apply the aforementioned law of exponents, carefully given that the term in the denominator of the fraction has a negative exponent. Therefore we used parentheses. We then simplified the expression in the exponent,

Next we'll recall the law of exponents for multiplication between terms with identical bases:

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

and we'll apply this law to the last expression that we obtained:

$9^{300}\cdot9^{252}\cdot9^{-549}=9^{300+252+(-549)}=9^{300+252-549}=9^3$

Let's summarize the steps so far:

$9^{300}\cdot\frac{1}{9^{-252}}\cdot9^{-549}=9^{300}\cdot9^{252}\cdot9^{-549} =9^{300+252-549}=9^3$

Note that there isn't such an answer among the answer choices, however we can always represent the expression that we obtained as a term with a negative exponent by taking the minus sign outside the parentheses in the exponent as follows:

$9^3=9^{-(-3)}$

We'll once again use the law of negative exponents:

$a^{-n}=\frac{1}{a^n}$

Let's apply it to the last expression that we obtained:

$9^3=9^{-(-3)}=\frac{1}{9^{-3}}$

Therefore :

$9^{300}\cdot\frac{1}{9^{-252}}\cdot9^{-549}=9^3 =9^{-(-3)}=\frac{1}{9^{-3}}$

The correct answer is answer A.

Apply the laws of exponents for negative exponents, in the opposite direction:

$\frac{1}{a^n} = a^{-n}$

Thus we can handle the leftmost term in the multiplication:

$\frac{1}{-3}\cdot3^{-4}\cdot5^3=-\frac{1}{3}\cdot3^{-4}\cdot5^3=-3^{-1}\cdot3^{-4}\cdot5^3$ In the first step we simplified the first fraction whilst remembering that dividing a positive number by a negative number gives a negative result. In the second step we applied the aforementioned law of exponents,

Before we continue, let's note and emphasize that the minus sign is not under the exponent in the first term of the multiplication, meaning - the exponent doesn't apply to it but only to the number 3.

Next, we'll recall the law of exponents for multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$and we'll apply this law to the last expression we got:

$-3^{-1}\cdot3^{-4}\cdot5^3=-3^{-1+(-4)}\cdot5^3=-3^{-1-4}\cdot5^3=-3^{-5}\cdot5^3$when we applied the aforementioned law of exponents only to the terms with identical bases and carried the minus sign throughout the calculation for the reason we mentioned earlier,

Let's summarize the steps so far:

$\frac{1}{-3}\cdot3^{-4}\cdot5^3=-3^{-1}\cdot3^{-4}\cdot5^3 =-3^{-1-4}\cdot5^3=-3^{-5}\cdot5^3$

Note that this answer isn't among the answer choices, however, we can apply the negative exponent law once again:

$a^{-n}=\frac{1}{a^n}$We'll apply it to the first term in the multiplication of terms that we obtained in the last step:

$-3^{-5}\cdot5^3=-\frac{1}{3^5}\cdot5^3=-\frac{5^3}{3^5}$In the first step we applied the aforementioned law to the first term in the multiplication, and in the next step we performed the fraction multiplication whilst remembering that multiplying by a fraction is essentially multiplying by the numerator,

Let's summarize the solution steps again:
$\frac{1}{-3}\cdot3^{-4}\cdot5^3=-3^{-1}\cdot3^{-4}\cdot5^3 =-3^{-5}\cdot5^3 =-\frac{5^3}{3^5}$

Therefore, the correct answer is answer B.