Powers of a Fraction: converting Negative Exponents to Positive Exponents

To simplify the expression $\left(\frac{1}{20}\right)^{-7}$, we will apply the rule for negative exponents. The key idea is that a negative exponent indicates taking the reciprocal and converting the exponent to a positive:

• Start with the expression: $\left(\frac{1}{20}\right)^{-7}$.
• Apply the negative exponent rule: $\left(\frac{1}{a}\right)^{-n} = a^n$.
• For our expression: $\left(\frac{1}{20}\right)^{-7}$ becomes $20^7$.

Therefore, $\left(\frac{1}{20}\right)^{-7}$ simplifies to $20^7$.

Thus, the correct answer is $20^7$.

To solve for $\left(\frac{1}{60}\right)^{-4}$, we apply the rule for negative exponents.

Step 1: Use the negative exponent rule: For any non-zero number $a$, $a^{-n} = \frac{1}{a^n}$. Thus,

$\left(\frac{1}{60}\right)^{-4} = \left(\frac{60}{1}\right)^4$.

Step 2: Simplify $\left(\frac{60}{1}\right)^4$ by recognizing the identity $\frac{60}{1} = 60$, so it follows that:

$\left(\frac{60}{1}\right)^4 = 60^4$.

Therefore, the simplified expression is $60^4$.

The correct answer is $60^4$

To solve the expression $\left(\frac{15}{21}\right)^{-3}$, we will apply the rule for converting negative exponents into positive exponents.

Step 1: Recognize that the negative exponent indicates the reciprocal of the base raised to the positive equivalent of the exponent. Thus, we use the formula:

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$.

Step 2: Apply this formula to our expression:

$\left(\frac{15}{21}\right)^{-3} = \left(\frac{21}{15}\right)^3$.

Therefore, the solution to the problem is $\left(\frac{21}{15}\right)^3$, which corresponds to choice 3.

To solve the problem, apply the negative exponent rule:

• For any fraction $\frac{a}{b}$ with a negative exponent $-n$, apply the rule: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$.

Apply this rule to the given expression $\left(\frac{10}{13}\right)^{-2}$:

$\left(\frac{10}{13}\right)^{-2} = \left(\frac{13}{10}\right)^{2}$

Therefore, the correct expression with a positive exponent is $\left(\frac{13}{10}\right)^{2}$.

In the provided choices, this is option:

• $\left(\frac{13}{10}\right)^2$

Hence, the correct expression is $\left(\frac{13}{10}\right)^2$.

To solve the problem of converting $\left(\frac{2}{5}\right)^{-2}$ to positive exponents, we use the rule for negative exponents:

Negative exponent rule states:

• $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ — This indicates that we invert the fraction and change the sign of the exponent to positive.

Given expression: $\left(\frac{2}{5}\right)^{-2}$.

Application: By using the rule, the negative exponent instructs us to reciprocate the fraction:

$\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2}$.

The positive exponent $(2)$ indicates the expression is squared. Thus, our action is complete with no further action required.

Thus, the correctly transformed expression of $\left(\frac{2}{5}\right)^{-2}$ is indeed:

$\left(\frac{5}{2}\right)^2$.

To solve the problem, we will follow these steps:

• Step 1: Identify the expression and apply the rule for negative exponents.
• Step 2: Take the reciprocal of the given fraction $\frac{10}{17}$.
• Step 3: Raise the reciprocal to the positive power of 5.

Now, let's work through each step:
Step 1: We start with the problem expression $\left(\frac{10}{17}\right)^{-5}$. According to the laws of exponents, a negative exponent means the reciprocal of the base should be raised to the positive of that exponent.
Step 2: Take the reciprocal of $\frac{10}{17}$, which is $\frac{17}{10}$.
Step 3: Raise the reciprocal $\frac{17}{10}$ to the power of 5, resulting in $\left(\frac{17}{10}\right)^5$.

Therefore, the equivalent expression is $\left(\frac{17}{10}\right)^5$.

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

• Step 1: Identify the given expression with a negative exponent.
• Step 2: Apply the rule for negative exponents, which allows us to convert the expression into a positive exponent form.
• Step 3: Perform the calculation of the new expression.

Now, let's work through each step:

Step 1: The expression given is $\left(\frac{1}{3}\right)^{-4}$, which involves a negative exponent.

Step 2: According to the exponent rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$, we can rewrite the expression with a positive exponent by inverting the fraction:

$\left(\frac{1}{3}\right)^{-4} = \left(\frac{3}{1}\right)^4 = 3^4$.

Step 3: Calculate $3^4$.

The calculation $3^4$ is as follows:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$.

However, since the problem specifically asks for the corresponding expression before calculation to numerical form, the answer remains $3^4$.

Therefore, the answer to the problem, in terms of an equivalent expression, is $3^4$.

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

• Step 1: Identify the given fraction and the negative exponent.
• Step 2: Apply the conversion rule from negative to positive exponents on fractions.

Now, let's work through each step:
Step 1: The given expression is $\left(\frac{3}{8}\right)^{-5}$. This indicates a fraction raised to a negative power.
Step 2: Applying the rule $(\frac{a}{b})^{-n} = (\frac{b}{a})^{n}$, we invert the fraction and change the exponent to positive. This gives us the expression $\left(\frac{8}{3}\right)^5$.

Therefore, the solution to the problem is $\left(\frac{8}{3}\right)^5$.

To solve this problem, we need to convert the expression $\left(\frac{5}{6}\right)^{-3}$ into a form with positive exponents.

The negative exponent rule states that $x^{-n} = \frac{1}{x^n}$. Applying this to our given fraction:

$\left(\frac{5}{6}\right)^{-3} = \left(\frac{6}{5}\right)^{3}$.

This means we take the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$, and then raise it to the power of 3.

Therefore, the correct expression is $\left(\frac{6}{5}\right)^3$.

This matches choice 1 in the list of possible answers.

We are given the expression: $\left(\frac{1}{2\times3\times4}\right)^{-2}$. We need to simplify it using the rules of exponents.

• Step 1: Identify the base of the exponent.
  The base is $\frac{1}{2\times3\times4}$.
  

• Step 2: Apply the rule for negative exponents.
  For a fraction $\frac{1}{a}$ with a negative exponent, $\left( \frac{1}{a} \right)^{-n} = a^n$. Therefore, $\left(\frac{1}{2\times3\times4}\right)^{-2} = (2\times3\times4)^2$.
  

• Step 3: Expand the expression.
  $(2\times3\times4)^2 = 2^2 \times 3^2 \times 4^2$.

Thus, the simplified expression is: $2^2\times3^2\times4^2$

We begin with the expression: $\frac{1}{a^{-x}}$.
Our goal is to simplify this expression while converting any negative exponents into positive ones.

• Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$.
• Correspondingly, $\frac{1}{a^{-n}} = a^n$.
• Thus, in our expression $\frac{1}{a^{-x}}$, the negative exponent can be converted and flipped to the numerator by the rule: $\frac{1}{a^{-x}} = a^x$.

Let's use the law of exponents for negative exponents:

$a^{-n} = \frac{1}{a^n}$and apply this law to the problem:

$10^8+10^{-4}+(\frac{1}{10})^{-16}=10^8+\frac{1}{10^4}+(10^{-1})^{-16}$$$when we apply the above law of exponents to the second term in the sum, and the same law but in the opposite direction - we'll apply it to the fraction inside the parentheses of the third term in the sum,

Now let's recall the law of exponents for exponent of an exponent:

$(a^m)^n=a^{m\cdot n}$we'll apply this law to the expression we got in the last step:

$10^8+\frac{1}{10^4}+(10^{-1})^{-16}=10^8+\frac{1}{10^4}+10^{(-1)\cdot(-16)}=10^8+\frac{1}{10^4}+10^{16}$when we apply this law to the third term from the left and then simplify the resulting expression,

Let's summarize the solution steps, we got that:

$10^8+10^{-4}+(\frac{1}{10})^{-16}=10^8+\frac{1}{10^4}+(10^{-1})^{-16} =10^8+\frac{1}{10^4}+10^{16}$Therefore the correct answer is answer A.

Let's start by emphasizing that this problem requires a different approach to applying the laws of exponents and is not as straightforward as many other problems solved so far. We should note that it's actually a very simplified expression, however, to understand which of the answers is correct, let's present it in a slightly different way,

Let's recall two of the laws of exponents:

a. The law of exponents raised to an exponent, but in the opposite direction:

$a^{m\cdot n} = (a^m)^n$b. The law of exponents applied to fractions, but in the opposite direction:

$\frac{a^n}{c^n} = \big(\frac{a}{c}\big)^n$

We'll work onthe two terms in the problem separately, starting with the first term on the left:

$\frac{z^{8n}}{m^{4t}}$

Note that both in the numerator and denominator, the number we are given in the exponents is a multiple of 4. Therefore, using the first law of exponents (in the opposite direction) mentioned above in a', we can represent both the term in the numerator and the term in the denominator as terms with an exponent of 4:

$\frac{z^{8n}}{m^{4t}}=\frac{z^{2n\cdot4}}{m^{t\cdot4}}=\frac{(z^{2n})^4}{(m^t)^4}$

First we see the exponents as a multiple of 4, and then we apply the law of exponents mentioned in a', to the numerator and denominator.

Next, we'll notice that both the numerator and the denominator are have the same exponent, and therefore we can use the second law of exponents mentioned in b', in the opposite direction:

$\frac{(z^{2n})^4}{(m^t)^4} =\big(\frac{z^{2n}}{m^t}\big)^4$

We could use the second law of exponents in its opposite direction because the terms in the numerator and denominator of the fraction have the same exponent.

Let's summarize the solution so far. We got that:

$\frac{z^{8n}}{m^{4t}}=\frac{(z^{2n})^4}{(m^t)^4}=\big(\frac{z^{2n}}{m^t}\big)^4$

Now let's stop here and take a look at the given answers:

Note that similar terms exist in all the answers, however, in answer a' the exponent (in this case its numerator and denominator are opposite to the expression we got in the last stage) is completely different from the exponent in the expression we got (that is - it's not even in the opposite sign to the exponent in the expression we got).

In addition, there's the coefficient 4 which doesn't exist in our expression, therefore we'll disqualify this answer,

Let's now refer to the proposed answer d' where only the first term from the multiplication in the given problem exists and it's clear that there's no information in the problem that could lead to the value of the second term in the multiplication being 1, so we'll disqualify this answer as well,

If so, we're left with answers b' or c', but the first term:

$(\frac{m^t}{z^{2n}})^{-4}$in them, is similar but not identical, to the term we got in the last stage:

$\big(\frac{z^{2n}}{m^t}\big)^4$The clear difference between them is in the exponent, which in the expression we got is positive and in answers b' and c' is negative,

This reminds us of the law of negative exponents:

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

Before we return to solving the problem let's understand this law in a slightly different, indirect way:

If we refer to this law as an equation (and it is indeed an equation for all intents and purposes), and multiply both sides of the equation by the common denominator which is:

$a^n$we'll get:

$a^{-n}=\frac{1}{a^n}\\ \frac{a^n}{1} =\frac{1}{a^n}\hspace{8pt} \text{/}\cdot a^n\\ a^n\cdot a^{-n}=1$

Here we remember that any number can be made into a fraction by writing it as itself divided by 1 , we applied this to the left side of the equation, then we multiplied by the common denominator and to know by how much we multiplied each numerator (after finding the common denominator) we asked the question "by how much did we multiply the current denominator to get the common denominator?".

Let's see the result we got:

$a^n\cdot a^{-n}=1$meaning that $a^n,\hspace{4pt}a^{-n}$are reciprocal numbers to each other, or in other words:

$a^n$is reciprocal to-$a^{-n}$(and vice versa).

We can apply this understanding to the problem if we also remember that the reciprocal number to a fraction is the number gotten by swapping the numerator and denominator, meaning that the fractions:

$\frac{a}{b},\hspace{4pt}\frac{b}{a}$

are reciprocal fractions to each other- which makes sense, since multiplying them will give us 1.

And if we combine this with the previous understanding, we can conclude that:

$\big(\frac{a}{b}\big)^{-1}=\frac{b}{a}$

meaning that raising a fraction to the power of negative one will give a result that is the reciprocal fraction, gotten by swapping the numerator and denominator.

Let's return to the problem and apply these understandings. First we'll briefly review what we've done already:

We dealt with the first term from the left from the problem:

$\frac{z^{8n}}{m^{4t}}\cdot c^z=\text{?}$and after dealing with it using the laws of exponents we got that it can be represented as:

$\frac{z^{8n}}{m^{4t}}=\frac{(z^{2n})^4}{(m^t)^4}=\big(\frac{z^{2n}}{m^t}\big)^4$

Then after disqualifying answers a' and d' for the reasons mentioned earlier, we wanted to show that the term we got in the last stage:

$\big(\frac{z^{2n}}{m^t}\big)^4$is identical to the first term in the multiplication of terms in answers b' -c':

$(\frac{m^t}{z^{2n}})^{-4}$Now after we understood that raising a fraction to the power of $-1$will swap between the numerator and denominator, meaning that:

$\big(\frac{a}{b}\big)^{-1}=\frac{b}{a}$we can return to the expression we got for the first term in the multiplication , and present it as a term with a negative exponent and in the denominator of the fraction:

$\big(\frac{z^{2n}}{m^t}\big)^4 = \big(\big(\frac{m^t }{z^{2n}}\big)^{-1}\big)^4 = \big(\frac{m^t }{z^{2n}}\big)^{-1\cdot4}=\big(\frac{m^t }{z^{2n}}\big)^{-4}$

We applied the aforementioned understanding inside the parentheses and presented the fraction as the reciprocal fraction to the power of $-1$and in the next stage we applied the law of exponents raised to an exponent:

$(a^m)^n=a^{m\cdot n}$to the expression we got, then we simplified the expression in the exponent,

If so, we proved that the expression we got in the last step (the first expression in the problem) is identical to the first expression in the multiplication in answers b' and c',

We'll continue then and focus the choosing between these options for the second term in the problem.

The second term in the multiplication in the problem is:

$c^z$

$$Let's return to the proposed answers b' and c' (which haven't been disqualified yet) and note that actually only the second term in the multiplication in answer b' is similar to this term (and not in answer c'), except that it's in the denominator and has a negative exponent while in our case (the term in the problem) it's in the numerator (see note at the end of solution) and has a positive exponent.

This will again remind us of the law of negative exponents, meaning we'll want to present the term in the problem we're currently dealing with, as having a negative exponent and in the denominator, we'll do this as follows:

$c^z=c^{-(\underline{-z})}=\frac{1}{c^{\underline{-z}}}$$$

Here we present the term in question as having a negative exponent , using the multiplication laws, and then we applied the law of negative exponents:

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

Carefully - because the expression we're dealing with now has a negative sign (indicated by an underline , both in the law of exponents here and in the last calculation made)

Let's summarize:

$\frac{z^{8n}}{m^{4t}}\cdot c^z=\big(\frac{z^{2n}}{m^t}\big)^4\cdot c^z=\big(\frac{m^t }{z^{2n}}\big)^{-4}\cdot\frac{1}{c^{-z}}$And therefore the correct answer is answer b'.

Note:

When we see "the number in the numerator" when there's no fraction, it's because we can always refer to any number as a number 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 by writing it like this:

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

We will use the following law of negative exponents:

$a^{-n}=\frac{1}{a^n}$Before we approach solving the problem we will understand this law in a slightly different way:

Note that if we treat this law as an equation (and it is indeed an equation in every sense), and multiply both sides of the equation by the common denominator which is:

$a^n$we get:

$a^{-n}=\frac{1}{a^n}\\ \frac{a^n}{1} =\frac{1}{a^n}\hspace{8pt} \text{/}\cdot a^n\\ a^n\cdot a^{-n}=1$In the first stage we remembered that any number can be made into a fraction simply by dividing by 1, we applied this to the left side of the equation, then we multiplied by the common denominator. To know by how much we need to multiply each numerator (after reduction with the common denominator) we ask the question "By how much did we multiply the original denominator to get the common denominator?".

Let's pay attention to the result we got:

$a^n\cdot a^{-n}=1$meaning that $a^n,\hspace{4pt}a^{-n}$are reciprocal numbers to each other, or in other words:

$a^n$is reciprocal to $a^{-n}$(and vice versa),

and in particular:

$a,\hspace{4pt}a^{-1}$are reciprocal to each other,

We can apply this understanding to the problem if we also remember that we get the reciprocal number of a fraction by swapping the numerator and denominator, meaning that the fractions:

$\frac{a}{b},\hspace{4pt}\frac{b}{a}$are reciprocal fractions to each other - which can be easily checked, since multiplying them will clearly give us 1.

If we combine this with the previous understanding, we can conclude that:

$\big(\frac{a}{b}\big)^{-1}=\frac{b}{a}$meaning that raising a fraction to the power of minus one will always give the reciprocal fraction, obtained by swapping the numerator and denominator.

Let's return to the problem and apply these understandings.

In addition we'll remember the law of multiplying exponents, but in the opposite direction:

$(a^m)^n=a^{m\cdot n}$We'll also apply this law to the problem:

$\big (\frac{7}{8} \big )^{-4}\cdot\frac{8}{7}+\frac{7}{8}\cdot \big (\frac{8}{7} \big )^{-3}=\text{?}$where we'll deal with each of the terms in the given sum separately:

a. We'll start by dealing separately with the first term from the left in the sum in the problem:

$\big (\frac{7}{8} \big )^{-4}\cdot\frac{8}{7}= \big (\frac{7}{8} \big )^{-1\cdot 4}\cdot\frac{8}{7}= \big (\big (\frac{7}{8}\big )^{-1} \big )^{4}\cdot\frac{8}{7}$In the first step, we presented the exponent expression as a multiplication, in the second step we applied the law of multiplying exponents in the opposite direction.

Next, we'll recall that raising a fraction to the power of negative one will always give the reciprocal fraction. We'll apply this to the first term in the multiplication expression we got in the last stage:

$\big (\big (\frac{7}{8}\big )^{-1} \big )^{4}\cdot\frac{8}{7}= \big (\frac{8}{7} \big )^{4}\cdot\frac{8}{7}$From here we notice that all the terms in the multiplication expression we got in the last stage have identical bases:

$\frac{8}{7}$So we'll remember the law of multiplying exponents with identical bases:$a^m\cdot a^n=a^{m+n}$and we'll apply this law to the expression we got in the last stage:

$\big (\frac{8}{7} \big )^{4}\cdot\frac{8}{7} =\big (\frac{8}{7} \big )^{4+1} =\big (\frac{8}{7} \big )^{5}$ In the first stage we applied the above-mentioned law of exponents while remembering that: $\frac{8}{7}=\big(\frac{8}{7}\big)^1$and in the following stages we simplified the expression in the exponent

Let's summarize what we've done so far:

For the first term in the sum in the problem, we got that:

$\big (\frac{7}{8} \big )^{-4}\cdot\frac{8}{7}= \big (\frac{8}{7} \big )^{4}\cdot\frac{8}{7} =\big (\frac{8}{7} \big )^{5}$

b. Now we'll move on to dealing with the second term:

$\frac{7}{8}\cdot \big (\frac{8}{7} \big )^{-3}$

We'll use the commutative law of multiplication and swap, for convenience, between the two terms in the multiplication expression we're dealing with now, then we'll apply (again) the law of multiplying exponents, but in its opposite direction:

$(a^m)^n=a^{m\cdot n}$ and then - we'll treat this term in the same way as we did for the first term.

$\big (\frac{8}{7} \big )^{-3}\cdot \frac{7}{8}= \big (\frac{8}{7} \big )^{-1\cdot 3}\cdot\frac{7}{8}= \big (\big (\frac{8}{7}\big )^{-1} \big )^{3}\cdot\frac{7}{8}$In the first step we present the exponent expression as a multiplication, in the second step we apply the law of multiplying exponents in its opposite direction.

Next, we'll apply the understanding that raising a fraction to the power of negative one will always give the reciprocal fraction.

We'll apply this to the first term in the multiplication expression we got in the last step:

$\big (\big (\frac{8}{7}\big )^{-1} \big )^{3}\cdot\frac{7}{8} = \big (\frac{7}{8} \big )^{3}\cdot\frac{7}{8}$From here we notice that all the terms in the multiplication expression we got in the last stage have identical bases:

$\frac{7}{8}$So we'll apply the law of multiplying exponents with identical bases:$a^m\cdot a^n=a^{m+n}$and we'll apply this law to the expression we got in the last stage:

$\big (\frac{7}{8} \big )^{3}\cdot\frac{7}{8} =\big (\frac{7}{8} \big )^{3+1} =\big (\frac{7}{8} \big )^{4}$In the first stage we applied the above-mentioned law of exponents while remembering that: $\frac{7}{8}=\big(\frac{7}{8}\big)^1$and in the following stages we simplified the expression in the exponent.

Let's summarize what we've got so far for the second term from the left in the problem.

We got that:

$\big (\frac{8}{7} \big )^{-3}\cdot \frac{7}{8} = \big (\frac{7}{8} \big )^{3}\cdot\frac{7}{8} =\big (\frac{7}{8} \big )^{4}$

Now let's return to the original problem and swap the original terms with solution a and b .

We got that:

$\big (\frac{7}{8} \big )^{-4}\cdot\frac{8}{7}+\frac{7}{8}\cdot \big (\frac{8}{7} \big )^{-3}= \big (\frac{8}{7} \big )^{4}\cdot\frac{8}{7} + \big (\frac{7}{8} \big )^{3}\cdot\frac{7}{8} =\big (\frac{8}{7} \big )^{5}+ \big (\frac{7}{8} \big )^{4}$

Therefore, the correct answer is answer a.