Negative Exponents: Calculating powers with negative exponents

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.

First, let's convert the decimal fraction in the problem to a simple fraction:

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

where we remembered that 0.25 is 25 hundredths, meaning:

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

If so, let's write the problem:

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

Now we'll use the negative exponent law:

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

and deal with the fraction expression inside the parentheses:

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

when we applied the above exponent law to the expression inside the parentheses,

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

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

and we'll apply this law to the expression we got in the last step:

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

where in the first step we carefully applied the above law and used parentheses in the exponent to perform the multiplication between the powers, then we simplified 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.

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 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:

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

We begin by using the negative exponent rule:

$b^{-n}=\frac{1}{b^n}$We apply it to the given expression and obtain the following:

$(3a)^{-2}=\frac{1}{(3a)^2}$We then use the power rule for parentheses:

$(x\cdot y)^n=x^n\cdot y^n$We apply it to the denominator of the expression and obtain the following:

$\frac{1}{(3a)^2}=\frac{1}{3^2a^2}=\frac{1}{9a^2}$Let's summarize the solution to the problem:

$(3a)^{-2}=\frac{1}{(3a)^2} =\frac{1}{9a^2}$

Therefore, the correct answer is option A.