Applying Combined Exponents Rules: Multiplying Exponents with the same base

We use the power property to multiply terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We apply it in the problem:

$Y^2+Y^6-Y^5\cdot Y=Y^2+Y^6-Y^{5+1}=Y^2+Y^6-Y^6=Y^2$When we apply the previous property to the third expression from the left in the sum, and then simplify the total expression by adding like terms.

Therefore, the correct answer is option D.

We begin by using the rule for multiplying exponents. (the multiplication between terms with identical bases):

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

$7^5\cdot7^{-6}=7^{5+(-6)}=7^{5-6}=7^{-1}$When in a first stage we begin by applying the aforementioned rule and then continue on to simplify the expression in the exponent,

Next, we use the negative exponent rule:

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

$7^{-1}=\frac{1}{7^1}=\frac{1}{7}$We then summarise the solution to the problem: $7^5\cdot7^{-6}=7^{-1}=\frac{1}{7}$Therefore, the correct answer is option B.

We begin by using the power rule of exponents; for the multiplication of terms with identical bases:

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

$12^4\cdot12^{-6}=12^{4+(-6)}=12^{4-6}=12^{-2}$When in a first stage we apply the aforementioned rule and then simplify the subsequent expression in the exponent,

Next, we use the negative exponent rule:

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

$12^{-2}=\frac{1}{12^2}=\frac{1}{144}$Lastly we summarise the solution to the problem: $12^4\cdot12^{-6}=12^{-2} =\frac{1}{144}$Therefore, the correct answer is option A.

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 power law for multiplication between terms with identical bases:

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

$\big(-\frac{1}{8}\big)^8\cdot\big(-\frac{1}{8}\big)^{-3}=\big(-\frac{1}{8}\big)^{8+(-3)}=\big(-\frac{1}{8}\big)^{8-3}=\big(-\frac{1}{8}\big)^5$

In the first stage we applied the above power law and in the following stages we simplified the expression in the exponent,

Let's continue and use the power law for power of terms in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$

We'll apply this law to the expression that we obtained in the last stage:

$\big(-\frac{1}{8}\big)^5=\big(-1\cdot\frac{1}{8}\big)^5=(-1)^5\cdot\big(\frac{1}{8}\big)^5=-1\cdot\big(\frac{1}{8}\big)^5=-\big(\frac{1}{8}\big)^5$

In the first stage we presented the expression in parentheses as a multiplication between negative one and a positive number. In the next stage we applied the above power law and then simplified the expression we obtained whilst noting that negative one to an odd power will (always) give the result negative one.

Next we'll recall two additional power laws:

a. The negative power law:

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

b. The power law for power of a power:

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

We'll continue and apply these two laws to the expression that we obtained in the last stage:

$-\big(\frac{1}{8}\big)^5=-(8^{-1})^5=-8^{(-1)\cdot5}=-8^{-5}$

In the first stage we presented the fraction inside the parentheses as a term with a negative power using the above power law for negative power mentioned in a. above. In the next stage we applied the power law for power of a power mentioned in b. above carefully, given that the term inside the parentheses has a negative power. We then simplified the expression in the exponent.

Let's summarize the solution :

$\big(-\frac{1}{8}\big)^8\cdot\big(-\frac{1}{8}\big)^{-3}=\big(-\frac{1}{8}\big)^5=-\big(\frac{1}{8}\big)^5=-(8^{-1})^5=-8^{-5}$

Therefore the correct answer is answer d.

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

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

to the middle term in the multiplication in the problem:

$4^{2x}\cdot\frac{1}{4}\cdot4^{-2}=4^{2x}\cdot4^{-1}\cdot4^{-2}$

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 that we obtained :

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

We obtained the most simplified expression,

Let's summarize the steps so far, as follows:

$4^{2x}\cdot\frac{1}{4}\cdot4^{-2}=4^{2x}\cdot4^{-1}\cdot4^{-2} =4^{2x-3}$

A quick look at the options will reveal that there isn't such an answer among the options and another check of what we've done so far will show that there are no calculation errors,

This means that another mathematical manipulation is needed on the expression we got, a hint for the required manipulation could be the fact that answer D is similar to our expression but the exponent has a minus sign compared to the exponent we got in the final expression and the expression itself is in a fraction where the numerator is 1, which reminds us of the negative exponent law, let's check this suspicion and handle the expression we got in the following way:

$4^{2x-3}=4^{-(-2x+3)}=4^{-(3+(-2x))}=4^{-(3-2x)}$

The goal is to present the expression that we obtained in the form of a term with a negative exponent. We did this by taking the minus sign outside the parentheses in the exponent and rearranging the expression inside the parentheses using the commutative law of addition and then simplified the expression in parentheses,

Now let's use the negative exponent law again:

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

And apply it to the expression that we obtained:

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

Therefore the expression that we obtained earlier can be written as:

$4^{2x}\cdot\frac{1}{4}\cdot4^{-2} =4^{2x-3} = 4^{-(3-2x)}=\frac{1}{4^{3-2x}}$

The correct answer is indeed answer D.

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.