Applying Combined Exponents Rules: Number of terms

$(3\times4\times5)^4=$

We use the power law for multiplication within parentheses:

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

$(3\cdot4\cdot5)^4=3^4\cdot4^4\cdot5^4$Therefore, the correct answer is option b.

Note:

From the formula of the power property mentioned above, we understand that it refers not only to two terms of the multiplication within parentheses, but also for multiple terms within parentheses.

$3^44^45^4$

$(3\times2\times4\times6)^{-4}=$

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.).

$3^{-4}\times2^{-4}\times4^{-4}\times6^{-4}$

$(4\times7\times3)^2=$

We use the power law for multiplication within parentheses:

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

$(4\cdot7\cdot3)^2=4^2\cdot7^2\cdot3^2$Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we understand that we can apply it not only to the multiplication of two terms within parentheses, but is also for multiple terms within parentheses.

$4^2\times7^2\times3^2$

$(2^2)^3+(3^3)^4+(9^2)^6=$

We use the formula:

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

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

$2^6+3^{12}+9^{12}$

$(2\times8\times7)^2=$

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 then apply the above rule to the problem:

$(2\cdot8\cdot7)^2=2^2\cdot8^2\cdot7^2$Therefore, the correct answer is option d.

Note:

From 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.).

$12,544$

$(7\cdot4\cdot6\cdot3)^4= \text{?}$

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.

$7^4\cdot4^4\cdot6^4\cdot3^4$

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

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.).

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

Solve the exercise:

$(x^2\times3)^2=$

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.

$9x^4$

$(y^3\times x^2)^4=$

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.

$y^{12}x^8$

$((8by)^3)^y+(3^x)^a=$

$\left(8by\right)^{3\cdot y}+3^{x\cdot a}$

We begin by applying the following rule:

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

We then open the parentheses according to the above rule.

$\left(abc\right)^x=a^x\cdot b^x\cdot c^x$

$8^{3y}\cdot b^{3y}\cdot y^{3y}+3^{xa}$

$8^{3y}\times b^{3y}\times y^{3y}+3^{ax}$

$(\frac{13}{2})^0\cdot(\frac{2}{13})^{-2}\cdot(\frac{13}{2})^{-5}=\text{?}$

$(\frac{2}{13})^3$

$((7\times3)^2)^6+(3^{-1})^3\times(2^3)^4=$

$21^{12}+3^{-3}\times2^{12}$

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

$x^{\frac{1}{2}}\times a^{\frac{3}{4}}$

Simplify the following:

$\lbrack\frac{a^4}{a^3}\times\frac{a^8}{a^7}\rbrack:\frac{a^{10}}{a^8}$

$1$

Simplify the following:

$\frac{a^{12}}{a^9}\times\frac{a^3}{a^4}=$

$a^2$

$((x^{\frac{1}{4}}\times3^2\times6^3)^{\frac{1}{4}})^8=$

$x^{\frac{1}{2}}\times3^4\times6^6$

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

$x^4y^6z^8$

$9^{300}\cdot\frac{1}{9^{-252}}\cdot9^{-549}=\text{?}$

$\frac{1}{9^{-3}}$

$\frac{1}{-3}\cdot3^{-4}\cdot5^3=\text{?}$

$-\frac{5^3}{3^5}$

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

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