Question

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Answer 1

QUESTION 1

$(y^{5})^2$

To simplify the above expression, we apply the laws of indices.

Recall that

$(a^m)^{2}=a^m \times a^m$

Therefore,

$(y^{5})^2=y^{5} \times y^{5}$

Now we apply the product rule of indices.

Recall again that;

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

$\Rightarrow (y^{5})^2=y^{5+5}$

$\Rightarrow (y^{5})^2=y^{10}$

QUICK SOLUTION

Recall that;

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

$\Rightarrow (y^{5})^2=y^{5\times 2}$

$\Rightarrow (y^{5})^2=y^{10}$

QUESTION 2

$(n^{7})^4$

To simplify the above expression, we apply the laws of indices.

Recall that;

$(a^m)^{4}=a^m \times a^m\times a^m\times a^m$

Therefore,

$(n^{7})^4=n^{7}\times n^{7}\times n^{7}\timesn^{7}$

Now we apply the product rule of indices.

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

$\Rightarrow (n^{7})^4=n^{7+7+7+7}$

$\Rightarrow (n^{7})^4=n^{28}$

QUICK SOLUTION

Recall that;

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

$\Rightarrow (n^{4})^4=n^{7\times 4}$

$\Rightarrow (n^{7})^4=n^{28}$

ANSWER TO QUESTION 3

$(x^2)^5(x^3)$

Let us use the law;

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

to simplify the first part first while maintaining the right part for now.

$\Rightarrow (x^2)^5(x^3)=(x^{2 \times 5})(x^3)$

$\Rightarrow (x^2)^5(x^3)=(x^{10})(x^3)$

Now we apply the product rule of indices.

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

$\Rightarrow (x^2)^5(x^3)=x^{10+3}$

$\Rightarrow (x^2)^5(x^3)=x^{13}$

ANSWER TO QUESTION 4

$-3(ab^4)^3$

We first share the index for each factor in the parenthesis.

$\Rightarrow -3(ab^4)^3=-3(a^3)(b^4)^3$

We now use the law,

$(a^m)^{n}=a^{(m\times n)}$ for the right most factor.

$\Rightarrow -3(ab^4)^3=-3(a^3)(b^{4\times 3})$

$\Rightarrow -3(ab^4)^3=-3a^3b^{12}$

ANSWER TO QUESTION 5

$(-3ab^4)^3$

We first share the index for each factor in the parenthesis.

$\Rightarrow (-3ab^4)^3=(-3)^3(a^3)(b^4)^3$

We now use the law,

$(a^m)^{n}=a^{(m\times n)}$ for the right most factor.

$\Rightarrow (-3ab^4)^3=(-3)^3(a^3)(b^{4\times 3})$

$\Rightarrow (-3ab^4)^3=-27a^3b^{12}$

ANSWER TO QUESTION 6

$(4x^2b)^3$

We  first share the exponent for each of the factors inside the parenthesis.

$(4x^2b)^3=4^3 (x^2)^3 b^3$

We now use the law,

$(a^m)^{n}=a^{(m\times n)}$ for the middle factor.

$(4x^2b)^3=4^3 \times x^{2 \times 3} b^3$

$(4x^2b)^3=64x^{6} b^3$

ANSWER TO QUESTION 7

$(4a^2)^2(b^3)$

We share the exponent for each factor in the parenthesis.

$(4a^2)^2(b^3)=(4)^2(a^2)^2(b^3)$

We now use the law,

$(a^m)^{n}=a^{(m\times n)}$ for the middle factor.

$(4a^2)^2(b^3)=(4)^2(a^{2\times 2}(b^3)$

$\Rightarrow (4a^2)^2(b^3)=16a^{4}b^3$

ANSWER TO QUESTION 8

$(4x)^2(b^3)$

We share the exponent for each factor in the parenthesis.

$(4x)^2(b^3)=(4)^2(x^2)(b^3)$

$\Rightarrow (4a^2)^2(b^3)=16x^{2}b^3$

ANSWER TO QUESTION 9

$(x^2y^4)^5$

We share the exponent for each factor in the parenthesis.

$(x^2y^4)^5=(x^2)^5(y^4)^5$

$(x^2y^4)^5=(x^2)^5(y^4)^5$

We now use the law,

$(a^m)^{n}=a^{(m\times n)}$ to simplify each factor.

$(x^2y^4)^5=(x^{2 \times 5})(y^{4 \times 5})$

$(x^2y^4)^5=(x^{10})(y^{20})$

$(x^2y^4)^5=x^{10}y^{20}$

ANSWER TO QUESTION 10

$(2a^3b^2)(b^3)^2$

Recall that,

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

$(2a^3b^2)(b^3)^2=(2a^3b^2)(b^{3 \times 2})$

$(2a^3b^2)(b^3)^2=(2a^3b^2)(b^{6})$

$(2a^3b^2)(b^3)^2=(2a^3b^2)(b^{6})$

We apply the product law to get,

$(2a^3b^2)(b^3)^2=2a^3b^{2+6}$

$(2a^3b^2)(b^3)^2=2a^3b^{8}$

ANSWER TO QUESTION 11

$(-4xy)^3(-2x^2)^3$

We share the index for each factor to get,

$(-4xy)^3(-2x^2)^3=(-4)^3(x^3)(y^3)(-2)^3(x^2)^3$

We simplify to get,

$(-4xy)^3(-2x^2)^3=-64x^3y^3\times -8x^6$

Applying the product rule gives,

$(-4xy)^3(-2x^2)^3=-64\times -8 x^{3+6}y^3$

$(-4xy)^3(-2x^2)^3=512x^{9}y^3$

ANSWER 12

$(-3j^2k^3)^2(2j^2k)^3$

We split the index for each factor.

$(-3j^2k^3)^2(2j^2k)^3=(-3)^2(j^2)^2(k^3)^2(2^3)(j^2)^3(k^3)$

We simplify to get,

$(-3j^2k^3)^2(2j^2k)^3=9\times 8(j^4)(k^6)(j^6)(k^3)$

$(-3j^2k^3)^2(2j^2k)^3=72j^{4+6}k^{6+3}$

$(-3j^2k^3)^2(2j^2k)^3=72j^{10}k^{9}$

ANSWER 13

$(25a^2b)^3(\frac{1}{5}abf)^2$

We share the index.

$(25a^2b)^3(\frac{1}{5}abf)^2=(25^3)(a^2)^3(b^3)(\frac{1}{5})^2a^2b^2f^2)$

$(25a^2b)^3(\frac{1}{5}abf)^2=(25^3)\times (\frac{1}{25}) (a^6)(b^3)a^2b^2f^2$

$(25a^2b)^3(\frac{1}{5}abf)^2=625 (a^{6+2}b^{3+2})$

$(25a^2b)^3(\frac{1}{5}abf)^2=625 (a^{8}b^{5})$

ANSWER 14.

$(2xy)^2(-3x^2)(4y^4)=2^2x^2y^2(-3x^2)(4y^4)$

$(2xy)^2(-3x^2)(4y^4)=4\times -3\times 4 x^{2+2}y^{2+4}$

$(2xy)^2(-3x^2)(4y^4)=-48x^{4}y^{6}$

SEE ATTACHMENT FOR CONTINUATION

Answer 2

Answer:

120= t875''''''''''

Step-by-step explanation: