Applying Combined Exponents Rules: Number of terms

We use the power property for an exponent that is applied to a set parentheses in which the terms are multiplied:

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

We apply the law in the problem:

$(7\cdot4\cdot6\cdot3)^4=7^4\cdot4^4\cdot6^4\cdot3^4$

When we apply the exponent to a parentheses with multiplication, we apply the exponent to each term of the multiplication separately, and we keep the multiplication between them.

Therefore, the correct answer is option a.

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.).

We begin by using the power rule for parentheses.

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

That is, the power applied to a product inside parentheses is applied to each of the terms within when the parentheses are opened,

We apply the above rule to the given problem:

$(3\cdot2\cdot4\cdot6)^{-4}=3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4}$

Therefore, the correct answer is option d.

Note:

According to the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

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

We have an exponent raised to another exponent with a multiplication between parentheses:

$(z\cdot t)^n=z^n\cdot t^n$This says that in a case where a power is applied to a multiplication between parentheses,the power is applied to each term of the multiplication when the parentheses are opened,

We apply it in the problem:

$(3x^2)^2=3^2(x^2)^2$With the second term of the multiplication we proceed carefully, since it is already in a power (that's why we use parentheses). The term will be raised using the power law for an exponent raised to another exponent:

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

$3^2(x^2)^2=9x^{2\cdot2}=9x^4$In the first step we raise the number to the power, and in the second step we multiply the exponent.

Therefore, the correct answer is option a.

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

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

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

Let's apply this rule to our problem:

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

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

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

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

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

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

where in the second stage we performed the multiplication in the exponents of the factors we obtained, while remembering that multiplying fractions means multiplying their numerators, and then - in the final stage, we simplified the fraction in the power of the first factor in the resulting product.

Therefore, the correct answer is answer A.

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.