Applying Combined Exponents Rules: Presenting powers with negative exponents as fractions

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will write the fraction in parentheses as a negative power with the help of the previously mentioned power:

$\frac{1}{4}=\frac{1}{4^1}=4^{-1}$We return to the problem, where we obtained:

$\big(\frac{1}{4}\big)^{-1}=(4^{-1})^{-1}$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$(4^{-1})^{-1}=4^{-1\cdot-1}=4^1=4$Therefore, the correct answer is option d.

We use the property of powers of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$5^{-2}=\frac{1}{5^2}=\frac{1}{25}$

Therefore, the correct answer is option d.

We use the power property of a negative exponent:

$a^{-n}=\frac{1}{a^n}$We will rewrite the fraction in parentheses as a negative power:

$\frac{1}{7}=7^{-1}$Let's return to the problem, where we had:

$\bigg( \big( \frac{1}{7}\big)^{-1}\bigg)^4=\big((7^{-1})^{-1} \big)^4$We continue and use the power property of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$And we apply it in the problem:

$\big((7^{-1})^{-1} \big)^4 =(7^{-1\cdot-1})^4=(7^1)^4=7^{1\cdot4}=7^4$Therefore, the correct answer is option c

We begin by using the power rule of negative exponents.

$a^{-n}=\frac{1}{a^n}$We then apply it to the problem:

$$$4^{-1}=\frac{1}{4^1}=\frac{1}{4}$We can therefore deduce that the correct answer is option B.

We begin by using the power rule of negative exponents.

$a^{-n}=\frac{1}{a^n}$We then apply it to the problem:

$$$2^{-5}=\frac{1}{2^5}=\frac{1}{32}$We can therefore deduce that the correct answer is option A.

We begin by using the power property for a negative exponent:

$b^{-n}=\frac{1}{b^n}$We apply it to the problem:

$(-7)^{-3}=\frac{1}{(-7)^3}$We then subsequently notice that each whole number inside the parentheses is raised to a negative power (that is, the number and its negative coefficient together) When using the previously mentioned power property: We are careful to take this into account,

We then continue by simplifying the expression in the denominator of the fraction, remembering the exponentiation property for the power of terms in multiplication:

$(a^m)^n=a^{m\cdot n}$We apply the resulting expression

$\frac{1}{(-7)^3}=\frac{1}{(-1\cdot7)^3}=\frac{1}{(-1)^3\cdot7^3}=\frac{1}{-1\cdot7^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

In summary we are able to deduce that the solution to the problem is as follows:

$(-7)^{-3}=\frac{1}{(-7)^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

$$$$Therefore, the correct answer is option B.

Using the rules of negative exponents: how to raise a number to a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the problem:

$$$$$7^{-24}=\frac{1}{7^{24}}$Therefore, the correct answer is option D.

In order to solve the exercise, we use the negative exponent rule.

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

We apply the rule to the given exercise:

$19^{-2}=\frac{1}{19^2}$

We can then continue and calculate the exponent.

$\frac{1}{19^2}=\frac{1}{361}$

We begin by using the negative exponent rule.

$b^{-n}=\frac{1}{b^n}$We apply it to the problem:

$$$$$a^{-4}=\frac{1}{a^4}$Therefore, the correct answer is option B.

We use the negative exponent rule.

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

We apply it to the problem in the opposite sense.:

$$$$$\frac{1}{8^3}=8^{-3}$

Therefore, the correct answer is option A.

Note that:

a.

$10=5\cdot2$

Recall the law of exponents for a multiplication operating inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

According to this, we obtain the following:

$(-5)^3=(-1\cdot5)^3=(-1)^3\cdot5^3=-1\cdot5^3=-5^3$

We want to use the knowledge in 'a' in order to obtain terms with identical bases in both the numerator and denominator,

Let's return to the problem and apply the knowledge obtained from both 'a' and 'b':

$\frac{10}{(-5)^3}=\frac{2\cdot5}{-5^3}=\frac{2}{-1}\cdot\frac{5}{5^3}=-2\cdot\frac{5}{5^3}$

In the first stage we used 'a' in the numerator and 'b' in the fraction's denominator. In the next stage we presented the fraction as a multiplication of fractions according to the rule for multiplying fractions, then we proceeded to simplify the first fraction in the multiplication.

Now we'll apply the law of exponents for division between terms with identical bases:

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

Apply this law to the expression as shown below:

$-2\cdot\frac{5}{5^3}=-2\cdot5^{1-3}=-2\cdot5^{-2}$

In the first stage we applied this law to the fraction in the multiplication and then proceeded to simplify the expression that we obtained,

Let's summarize the various steps of the solution:

$\frac{10}{(-5)^3} =-2\cdot\frac{5}{5^3} =-2\cdot5^{-2}$

Therefore, the correct answer is answer b.

To begin with we deal with the expression in the denominator of the fraction. Making note of the power rule for exponents (raising an exponent to another exponent):

$(a^m)^n=a^{m\cdot n}$We obtain the following:

$(-2)^7=(-1\cdot2)^7=(-1)^7\cdot2^7=-1\cdot2^7=-2^7$

We then return to the initial problem and apply the above information:

$\frac{1}{(-2)^7}=\frac{1}{-2^7}=\frac{1}{-1}\cdot\frac{1}{2^7}=-\frac{1}{2^7}$$$

In the last step we remember that:

$\frac{1}{-1}=-1$

Next, we remember the Negative Exponent rule ( raising exponents to a negative power)

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

$-\frac{1}{2^7}=-2^{-7}$Let's summarize the steps of the solution:

$\frac{1}{(-2)^7}=-\frac{1}{2^7} = -2^{-7}$

$$$$Therefore, the correct answer is option C.

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the given expression:

$\frac{1}{2^9}=2^{-9}$

Therefore, the correct answer is option A.

To begin with, we must remind ourselves of the Negative Exponent rule:

$a^{-n}=\frac{1}{a^n}$We apply it to the given expression :

$\frac{1}{12^3}=12^{-3}$Therefore, the correct answer is option A.

First let's recall the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$We'll apply it to the expression we received:

$\frac{1}{4^{-3}}=4^{-(-3)}=4^3=64$In the first stage, we carefully applied the above exponent rule, and since the term in the denominator is already a negative exponent, when using the mentioned rule we put the exponent of the term that was in the denominator in parentheses (this is to apply the minus sign associated with the exponent rule later), then we simplified the exponent expression that was obtained.

In the final stage, we calculated the actual numerical result of the expression we received.

Therefore, the correct answer is answer B.

Begin by converting the decimal fraction in the problem to a simple fraction:

$0.25=\frac{25}{100}=\frac{1}{4}$

Remember that 0.25 is 25 hundredths, meaning:

$25\cdot\frac{1}{100}=\frac{25}{100}$

Proceed to write the problem:

$(0.25)^{-2}=\big(\frac{1}{4}\big)^{-2}=\text{?}$

Apply the negative exponent law:

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

and proceed to deal with the fraction expression inside of the parentheses:

$\big(\frac{1}{4}\big)^{-2}=(4^{-1})^{-2}$

We applied the above exponent law to the expression inside of the parentheses.

Next, recall the power of a power law:

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

Apply this law to the expression that we obtained in the last step:

$(4^{-1})^{-2}=4^{(-1)\cdot(-2)}=4^2=16$

in the first step we carefully applied the above law and used parentheses in the exponent to perform the multiplication between the powers. We then proceeded to simplify the resulting expression, and finally calculated the numerical result from the last step.

Let's summarize the solution steps:

$(0.25)^{-2}=\big(\frac{1}{4}\big)^{-2}=4^{(-1)\cdot(-2)}=16$

Therefore, the correct answer is answer B.

We must first remind ourselves of the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$When applied to given the expression we obtain the following:

$7^{-4}=\frac{1}{7^4}=\frac{1}{2401}$

Therefore, the correct answer is option C.

First let's recall the negative exponent rule:

$b^{-n}=\frac{1}{b^n}$We'll apply it to the expression we received:

$(-5)^{-3}=\frac{1}{(-5)^3}$Next let's recall the power rule for expressions in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$And we'll apply it to the denominator of the expression we received:

$\frac{1}{(-5)^3}=\frac{1}{(-1\cdot5)^3}=\frac{1}{(-1)^3\cdot5^3}=\frac{1}{-1\cdot5^3}=-\frac{1}{5^3}=-\frac{1}{125}$In the first step, we expressed the negative number inside the parentheses in the denominator as a multiplication between a positive number and negative one, and then we used the power rule for expressions in parentheses to expand the parentheses, and then we simplified the expression.

Let's summarize the solution to the problem:

$(-5)^{-3}=\frac{1}{(-5)^3} =\frac{1}{-5^3}=-\frac{1}{125}$

Therefore, the correct answer is answer B.

We use the formula:

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

Therefore, we obtain:

$(\frac{3}{2})^4$

We use the formula:

$(\frac{b}{a})^n=\frac{b^n}{a^n}$

Therefore, we obtain:

$\frac{3^4}{2^4}=\frac{3\times3\times3\times3}{2\times2\times2\times2}=\frac{81}{16}$

First, let's recall the negative exponent rule:

$b^{-n}=\frac{1}{b^n}$We'll apply it to the expression we received:

$10^{-5}=\frac{1}{10^5}=\frac{1}{100000}=0.00001$In the final steps, we performed the exponentiation in the numerator and then wrote the answer as a decimal.

Therefore, the correct answer is option A.