Applying Combined Exponents Rules: Variables in the exponent of the power

$((14^{3x})^{2y})^{5a}=$

Using the power rule for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply the rule to the given problem:

$((14^{3x})^{2y})^{5a}=(14^{3x})^{2y\cdot5a}=14^{3x\cdot2y\cdot5a}=14^{30xya}$In the first step we applied the aforementioned power rule and removed the outer parentheses. In the next step we again applied the power rule and removed the remaining parentheses.

In the final step we simplified the resulting expression,

Therefore, through the rule of substitution (which is applied to the exponent of the power in the obtained expression) it can be concluded that the correct answer is answer D.

$14^{30axy}$

$2^{2x+1}\cdot2^5\cdot2^{3x}=$

Since the bases are the same, the exponents can be added:

$2x+1+5+3x=5x+6$

$2^{5x+6}$

$(2\cdot4\cdot8)^{a+3}=$

Let's begin by using the distributing exponents rule (An exponent outside of a parentheses needs to be distributed across all the numbers and variables within the parentheses)

$(x\cdot y)^n=x^n\cdot y^n$We first apply this rule to the given problem:

$(2\cdot4\cdot8)^{a+3}= 2^{a+3}4^{a+3}8^{a+3}$When then we apply the power to each of the terms of the product inside the parentheses separately and maintain the multiplication.

The correct answer is option d.

$2^{a+3}4^{a+3}8^{a+3}$

$((3^9)^{4x)^{5y}}=$

We use the power rule for an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$We apply this rule to the given problem:

$((3^9)^{4x})^{5y}= (3^9)^{4x\cdot 5y} =3^{9\cdot4x\cdot 5y}=3^{180xy}$In the first step we applied the previously mentioned power rule and removed the outer parentheses. In the next step we applied the power rule once again and removed the remaining parentheses. In the final step we simplified the resulting expression.

Therefore, the correct answer is option b.

$3^{180xy}$

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6=$

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

$a^m\cdot a^n=a^{m+n}$We apply the property for this problem:

$4^{2y}\cdot4^{-5}\cdot4^{-y}\cdot4^6= 4^{2y+(-5)+(-y)+6}=4^{2y-5-y+6}$We simplify the expression we got in the last step:

$4^{2y-5-y+6} =4^{y+1}$When we add similar terms in the exponent.

Therefore, the correct answer is option c.

$4^{y+1}$

Solve the following exercise:

$(4\times9\times11)^a$

We use the power law for a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$That is, a power applied to a multiplication between parentheses is applied to each term when the parentheses are opened,

We apply it in the problem:

$(4\cdot9\cdot11)^a=4^a\cdot9^a\cdot11^a$Therefore, the correct answer is option b.

Note:

From the power property formula mentioned, we can understand that it works not only with two terms of the multiplication between parentheses, but also valid with a multiplication between multiple terms in parentheses. As we can see in this problem.

$4^a9^a11^a$

$(4^x)^y=$

Using the law of powers for an exponent raised to another exponent:

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

$(4^x)^y=4^{xy}$Therefore, the correct answer is option a.

$4^{xy}$

Simplify:

$(5\cdot12\cdot4\cdot6)^{a+3bx}$

Use the power property for a power in parentheses where there is a multiplication of its terms:

$(x\cdot y)^n=x^n\cdot y^n$We apply this law to the problem expression:

$(5\cdot12\cdot4\cdot6)^{a+3bx}=5^{a+3bx}12^{a+3bx}4^{a+3bx}6^{a+3bx}$When we apply a power to parentheses where its terms are multiplied, we do it separately and keep the multiplication.

Therefore, the correct answer is option d.

$5^{a+3bx}12^{a+3bx}4^{a+3bx}6^{a+3bx}$

$7^{2x+1}\cdot7^{-1}\cdot7^x=$

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

$a^m\cdot a^n=a^{m+n}$We apply the property to our expression:

$7^{2x+1}\cdot7^{-1}\cdot7^x=7^{2x+1+(-1)+x}=7^{2x+1-1+x}$We simplify the expression we got in the last step:

$7^{2x+1-1+x}=7^{3x}$When we add similar terms in the exponent.

Therefore, the correct answer is option d.

$7^{3x}$

Simplify:

$(2\cdot3\cdot7\cdot9)^{ab+3}$

We begin by using the distributive law of exponents.

$(x\cdot y)^n=x^n\cdot y^n$We apply this property to the given problem :

$(2\cdot3\cdot7\cdot9)^{ab+3} =2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}$When we apply the power of parentheses to each of the terms of the product inside the parentheses separately and maintain the multiplication.

Therefore, the correct answer is option a.

$2^{ab+3}3^{ab+3}7^{ab+3}9^{ab+3}$

$3^x\cdot2^x\cdot3^{2x}=$

In this case we have 2 different bases, so we will add what can be added, that is, the exponents of $3$

$3^x\cdot2^x\cdot3^{2x}=2^x\cdot3^{3x}$

$3^{3x}\cdot2^x$

$((4x)^{3y})^2=$

$4^{6y}\cdot x^{6y}$

Solve for a:

$\frac{a^{3b}}{a^{2b}}\times a^b=$

$a^{2b}$

Solve the exercise:

$\frac{a^{2x}}{a^y}\times\frac{a^{2y}}{a^{-y}}=$

$a^{2(x+y)}$