Power of a Power: Variables in the exponent of the power

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.

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.

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.

We'll use the power rule for powers:

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

$((4x)^{3y})^2= (4x)^{3y\cdot2}=(4x)^{6y}$When in the first stage we applied the mentioned power rule and eliminated the outer parentheses, in the next stage we simplified the resulting expression,

Next, we'll recall the power rule for powers that applies to parentheses containing a product of terms:

$(a\cdot b)^n=a^n\cdot b^n$We'll apply this rule to the expression we got in the last stage:

$(4x)^{6y} =4^{6y}\cdot x^{6y}$When we applied the power to the parentheses to each term of the product inside the parentheses.

Therefore, the correct answer is answer D.