Power of a Power: Number of terms

We use the formula:

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

$(2^2)^3+(3^3)^4+(9^2)^6=2^{2\times3}+3^{3\times4}+9^{2\times6}=2^6+3^{12}+9^{12}$

Let's handle each expression in the problem separately:

a. We'll start with the leftmost expression, first calculating the result of the multiplication in parentheses, and then use the power rule for power to a power:

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

Let's apply this to the problem for the first expression from the left:

$((7\cdot3)^2)^6=(21^2)^6=21^{2\cdot6}=21^{12}$

where in the final step we calculated the result of multiplication in the power expression,

We're done with this expression, let's move on to the next expression from the left.

b. Let's continue with the second expression from the left, using the power rule for power to a power that we mentioned above and apply it separately to each factor in this expression:

$(3^{-1})^3\cdot(2^3)^4=3^{-1\cdot3}\cdot2^{3\cdot4}=3^{-3}\cdot2^{12}$

Note that the multiplication factors we got have different bases, so we cannot further simplify this expression,

Therefore, let's combine parts a and b above in the result of the original problem:

$((7\cdot3)^2)^6+(3^{-1})^3\cdot(2^3)^4=21^{12}+3^{-3}\cdot2^{12}$

Therefore, the correct answer is answer d.

Let's solve this in two stages. In the first stage, we'll use the power law for power of a product in parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

which states that when raising a product in parentheses to a power, the power applies to each factor of the product when opening the parentheses,

Let's apply this law to our problem:

$(x^2\cdot y^3\cdot z^4)^2=(x^2)^2\cdot(y^3)^2\cdot(z^4)^2$

where when opening the parentheses, we applied the power to each factor of the product separately, but since each of these factors is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power law for power of a power:

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

Let's apply this law to the expression we got:

$(x^2)^2\cdot(y^3)^2\cdot(z^4)^2=x^{2\cdot2}\cdot y^{3\cdot2}\cdot z^{4\cdot2}=x^4\cdot y^6\cdot z^8$

where in the second stage we performed the multiplication operation in the power exponents of the factors we obtained.

Therefore, the correct answer is answer D.

Note:

From the above formulation of the power law for parentheses, it might seem that it only refers to two factors in a product within parentheses, but in fact, it is valid for a power of a product of multiple factors in parentheses, as demonstrated in this problem and others,

It would be a good exercise to prove that if the above law is valid for a power of a product of two factors in parentheses (as it is formulated above), then it is also valid for a power of a product of multiple factors in parentheses (for example - three factors, etc.).

Let's solve this in two stages. In the first stage, we'll use the power rule for powers in parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

which states that when a power is applied to terms in parentheses, it applies to each term inside the parentheses when they are opened,

Let's apply this rule to our problem:

$\big((x^{\frac{1}{4}}\cdot3^2\cdot6^3)^{\frac{1}{4}}\big)^8=((x^{\frac{1}{4}})^{\frac{1}{4}}\cdot(3^2)^{\frac{1}{4}}\cdot(6^3)^{\frac{1}{4}})^8$

where when opening the parentheses, we applied the power to each term separately, but since each of these terms is raised to a power, we did this carefully and used parentheses,

Next, we'll use the power rule for a power raised to a power:

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

Let's apply this rule to the expression we got:

$(x^{\frac{1}{4}\cdot\frac{1}{4}}\cdot3^{2\cdot\frac{1}{4}}\cdot6^{3\cdot\frac{1}{4}})^8=(x^{\frac{1}{16}}\cdot3^{\frac{2}{4}}\cdot6^{\frac{3}{4}})^8=x^{\frac{1}{16}\cdot8}\cdot3^{\frac{2}{4}\cdot8}\cdot6^{\frac{3}{4}\cdot8}=x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}$

where in the second stage we performed multiplication in the fractions of the power expressions of the terms we obtained, remembering that multiplication in fractions is actually multiplication in the numerator, and then - in the final stage we simplified the fractions in the power expressions of the multiplication terms we got:

$x^{\frac{8}{16}}\cdot3^{\frac{16}{4}}\cdot6^{\frac{24}{4}}=x^{\frac{1}{2}}\cdot3^4\cdot6^6$

Therefore, the correct answer is answer B.

We will solve the problem in two steps, in the first step we will use the power of a product rule:

$(z\cdot t)^n=z^n\cdot t^n$The rule states that the power affecting a product within parentheses applies to each of the elements of the product when the parentheses are opened,

We begin by applying the law to the given problem:

$(y^3\cdot x^2)^4=(y^3)^4\cdot(x^2)^4$When we open the parentheses, we apply the power to each of the terms of the product separately, but since each of these terms is already raised to a power, we must be careful to use parentheses.

We then use the power of a power rule.

$(b^m)^n=b^{m\cdot n}$We apply the rule to the given problem and we should obtain the following result:

$(y^3)^4\cdot(x^2)^4=y^{3\cdot4}\cdot x^{2\cdot4}=y^{12}\cdot x^8$When in the second step we perform the multiplication operation on the power exponents of the obtained terms.

Therefore, the correct answer is option d.

In solving the problem we will use two laws of exponents, let's recall them:

a. The law of exponents for multiplying terms with identical bases:

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

b. The law of exponents for power of a power:

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

We will apply these two laws of exponents to the expression in the problem in two stages:

We'll start by applying the law of exponents mentioned in a to the second term from the left in the expression:

$9^{-3}\cdot9^4=9^{-3+4}=9^1=9$

After we applied the law of exponents mentioned in a in the first stage and simplified the resulting expression,

We'll continue to the next stage and apply the law of exponents mentioned in b and deal with the third term from the left in the expression, we'll do this in two steps:

$((7^2)^5)^6=(7^2)^{5\cdot6}=7^{2\cdot5\cdot6}=7^{60}$

When in the first step we applied the law of exponents mentioned in b and eliminated the outer parentheses, in the next step we applied the same law of exponents again and eliminated the remaining parentheses, in the following steps we simplified the resulting expression,

Let's summarize the two stages detailed above for the complete solution of the problem:

$(9\cdot7\cdot6)^3+9^{-3}\cdot9^4+((7^2)^5)^6+2^4 = (9\cdot7\cdot6)^3+9+7^{60}+2^4$

In the next step we'll calculate the result of multiplying the terms inside the parentheses in the leftmost term:

$(9\cdot7\cdot6)^3+9+7^{60}+2^4 =378^3+9+7^{60}+2^4$

Therefore the correct answer is answer d.

We'll use the power rule for a power:

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

We'll apply this rule to the expression in the problem in two stages:

$((9xyz)^3)^4+(a^y)^x= (9xyz)^{3\cdot4}+(a^y)^x=(9xyz)^{12}+a^{yx}$

In the first stage, for good order, we applied the above power rule first to the first term in the expression and dealt with the outer parentheses, then we simplified the expression in the exponent while simultaneously applying the mentioned power rule to the second term in the sum in the problem's expression,

We'll continue and recall the power rule for powers that applies to parentheses containing multiplication of terms:

$(w\cdot t)^n=w^n\cdot t^n$

We'll apply this power rule to the expression we got in the last stage:

$(9xyz)^{12}+(a^y)^x =9^{12} x^{12} y^{12}z^{12}+a^{yx}$

When we applied the mentioned power rule to the first term in the sum in the expression we got in the last stage, and applied the power on the parentheses to each of the multiplication terms inside the parentheses,

Let's summarize the solution steps so far, we got that:

$((9xyz)^3)^4+(a^y)^x=(9xyz)^{12}+a^{yx} =9^{12} x^{12} y^{12} z^{12}+a^{yx}$

Therefore the correct answer is answer D.