Applying Combined Exponents Rules: Using the laws of exponents

$7^4\cdot8^3\cdot(\frac{1}{7})^4=\text{?}$

We use the formula:

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

We decompose the fraction inside of the parentheses:

$(\frac{1}{7})^4=\frac{1^4}{7^4}$

We obtain:

$7^4\times8^3\times\frac{1^4}{7^4}$

We simplify the powers: $7^4$

We obtain:

$8^3\times1^4$

Remember that the number 1 in any power is equal to 1, thus we obtain:

$8^3\times1=8^3$

$8^3$

$(8\times9\times5\times3)^{-2}=$

We begin by applying the power rule to the products within the parentheses:

$(z\cdot t)^n=z^n\cdot t^n$That is, the power applied to a product within parentheses is applied to each of the terms when the parentheses are opened,

We apply the rule to the given problem:

$(8\cdot9\cdot5\cdot3)^{-2}=8^{-2}\cdot9^{-2}\cdot5^{-2}\cdot3^{-2}$Therefore, the correct answer is option c.

Note:

Whilst it could be understood that the above power rule applies only to two terms of the product within parentheses, in reality, it is also valid for the power over a multiplication of multiple terms within parentheses, as was seen in the above problem.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms within parentheses (as formulated above), then it is also valid for a power over several terms of the product within parentheses (for example - three terms, etc.).

$8^{-2}\times9^{-2}\times5^{-2}\times3^{-2}$

$(\frac{2}{3})^{-4}=\text{?}$

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:

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

$\frac{81}{16}$

Solve the exercise:

$Y^2+Y^6-Y^5\cdot Y=$

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.

$Y^2$

$\frac{9\cdot3}{8^0}=\text{?}$

We use the formula:

$a^0=1$

$\frac{9\times3}{8^0}=\frac{9\times3}{1}=9\times3$

We know that:

$9=3^2$

Therefore, we obtain:

$3^2\times3=3^2\times3^1$

We use the formula:

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

$3^2\times3^1=3^{2+1}=3^3$

$3^3$

Solve the exercise:

$a^2:a+a^3\cdot a^5=$

First we rewrite the first expression on the left of the problem as a fraction:

$\frac{a^2}{a}+a^3\cdot a^5$Then we use two properties of exponentiation, to multiply and divide terms with identical bases:

A.

$b^m\cdot b^n=b^{m+n}$2.

$\frac{b^m}{b^n}=b^{m-n}$Returning to the problem and applying the two properties of exponentiation mentioned earlier:

$\frac{a^2}{a}+a^3\cdot a^5=a^{2-1}+a^{3+5}=a^1+a^8=a+a^8$

Later on, keep in mind that we need to factor the expression we obtained in the last step by extracting the common factor,

Therefore, we extract from outside the parentheses the greatest common divisor to the two terms which are:

$a$ We obtain the expression:

$a+a^8=a(1+a^7)$when we use the property of exponentiation mentioned earlier in A.

$$$a^8=a^{1+7}=a^1\cdot a^7=a\cdot a^7$

Summarizing the solution to the problem and all the steps, we obtained the following:

$\frac{a^2}{a}+a^3\cdot a^5=a(1+a^{7)}$Therefore, the correct answer is option b.

$a(1+a^7)$

$5^{-3}\cdot5^0\cdot5^2\cdot5^5=$

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

$a^m\cdot a^n=a^{m+n}$Keep in mind that this property is also valid for several terms in the multiplication and not just for two, for example for the multiplication of three terms with the same base we obtain:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$When we use the mentioned power property twice, we could also perform the same calculation for four terms of the multiplication of five, etc.,

Let's return to the problem:

$$$$Keep in mind that all the terms of the multiplication have the same base, so we will use the previous property:

$5^{-3}\cdot5^0\cdot5^2\cdot5^5=5^{-3+0+2+5}=5^4$Therefore, the correct answer is option c.

Note:

Keep in mind that $5^0=1$

$5^4$

$4^5-4^6\cdot\frac{1}{4}=\text{?}$

0

$5^3+5^{-3}\cdot5^3=\text{?}$

$5^3+1$

$\frac{4^0\cdot6^7}{36^4\cdot9^0}=\text{?}$

$\frac{1}{6}$

$\frac{9^2\cdot3^{-4}}{6^3}=\text{?}$

$6^{-3}$

$\frac{2^0\cdot3^{-4}}{5^4\cdot9^2}=\text{?}$

$\frac{1}{5^4\cdot3^8}$

$\sqrt[6]{b^{12}}\cdot(\frac{1}{b})^2\cdot a=\text{?}$

$a$

$3^{-3}\cdot\frac{19^{35}\cdot19^{-32}}{19^3}=\text{?}$

$\frac{1}{27}$

$\frac{(-3)^5\cdot8^4}{(-3)^3(-3)^2(-3)^{-5}}=\text{?}$

$-3^5\cdot8^4$

$\frac{10^4\cdot0.1^{-3}\cdot10^{-8}}{1000}=\text{?}$

$0.0001$

$\frac{2^{-4}\cdot(\frac{1}{2})^8\cdot2^{10}}{2^3}=\text{?}$

$2^{-5}$

$9^4\cdot3^{-8}\cdot\frac{1}{3}=\text{?}$

$3^{-1}$

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

$5(5^3-1)$

$45^{-80}\cdot\frac{1}{45^{-81}}\cdot49\cdot7^{-5}=\text{?}$

$\frac{45}{7^3}$