QUESTION 1
[tex](y^{5})^2[/tex]
To simplify the above expression, we apply the laws of indices.
Recall that
[tex](a^m)^{2}=a^m \times a^m[/tex]
Therefore,
[tex](y^{5})^2=y^{5} \times y^{5}[/tex]
Now we apply the product rule of indices.
Recall again that;
[tex]a^m \times a^n=a^{m+n}[/tex]
[tex]\Rightarrow (y^{5})^2=y^{5+5}[/tex]
[tex]\Rightarrow (y^{5})^2=y^{10}[/tex]
QUICK SOLUTION
Recall that;
[tex](a^m)^{n}=a^(m\times n) [/tex]
[tex]\Rightarrow (y^{5})^2=y^{5\times 2}[/tex]
[tex]\Rightarrow (y^{5})^2=y^{10}[/tex]
QUESTION 2
[tex](n^{7})^4[/tex]
To simplify the above expression, we apply the laws of indices.
Recall that;
[tex](a^m)^{4}=a^m \times a^m\times a^m\times a^m[/tex]
Therefore,
[tex](n^{7})^4=n^{7}\times n^{7}\times n^{7}\timesn^{7} [/tex]
Now we apply the product rule of indices.
[tex]a^m \times a^n=a^{m+n}[/tex]
[tex]\Rightarrow (n^{7})^4=n^{7+7+7+7}[/tex]
[tex]\Rightarrow (n^{7})^4=n^{28}[/tex]
QUICK SOLUTION
Recall that;
[tex](a^m)^{n}=a^{(m\times n)}[/tex]
[tex]\Rightarrow (n^{4})^4=n^{7\times 4}[/tex]
[tex]\Rightarrow (n^{7})^4=n^{28}[/tex]
ANSWER TO QUESTION 3
[tex](x^2)^5(x^3)[/tex]
Let us use the law;
[tex](a^m)^{n}=a^{(m\times n)}[/tex]
to simplify the first part first while maintaining the right part for now.
[tex]\Rightarrow (x^2)^5(x^3)=(x^{2 \times 5})(x^3)[/tex]
[tex]\Rightarrow (x^2)^5(x^3)=(x^{10})(x^3)[/tex]
Now we apply the product rule of indices.
[tex]a^m \times a^n=a^{m+n}[/tex]
[tex]\Rightarrow (x^2)^5(x^3)=x^{10+3}[/tex]
[tex]\Rightarrow (x^2)^5(x^3)=x^{13}[/tex]
ANSWER TO QUESTION 4
[tex]-3(ab^4)^3[/tex]
We first share the index for each factor in the parenthesis.
[tex]\Rightarrow -3(ab^4)^3=-3(a^3)(b^4)^3[/tex]
We now use the law,
[tex](a^m)^{n}=a^{(m\times n)}[/tex] for the right most factor.
[tex]\Rightarrow -3(ab^4)^3=-3(a^3)(b^{4\times 3})[/tex]
[tex]\Rightarrow -3(ab^4)^3=-3a^3b^{12}[/tex]
ANSWER TO QUESTION 5
[tex](-3ab^4)^3[/tex]
We first share the index for each factor in the parenthesis.
[tex]\Rightarrow (-3ab^4)^3=(-3)^3(a^3)(b^4)^3[/tex]
We now use the law,
[tex](a^m)^{n}=a^{(m\times n)}[/tex] for the right most factor.
[tex]\Rightarrow (-3ab^4)^3=(-3)^3(a^3)(b^{4\times 3})[/tex]
[tex]\Rightarrow (-3ab^4)^3=-27a^3b^{12}[/tex]
ANSWER TO QUESTION 6
[tex](4x^2b)^3[/tex]
We first share the exponent for each of the factors inside the parenthesis.
[tex](4x^2b)^3=4^3 (x^2)^3 b^3[/tex]
We now use the law,
[tex](a^m)^{n}=a^{(m\times n)}[/tex] for the middle factor.
[tex](4x^2b)^3=4^3 \times x^{2 \times 3} b^3[/tex]
[tex](4x^2b)^3=64x^{6} b^3[/tex]
ANSWER TO QUESTION 7
[tex](4a^2)^2(b^3)[/tex]
We share the exponent for each factor in the parenthesis.
[tex](4a^2)^2(b^3)=(4)^2(a^2)^2(b^3)[/tex]
We now use the law,
[tex](a^m)^{n}=a^{(m\times n)}[/tex] for the middle factor.
[tex](4a^2)^2(b^3)=(4)^2(a^{2\times 2}(b^3)[/tex]
[tex]\Rightarrow (4a^2)^2(b^3)=16a^{4}b^3[/tex]
ANSWER TO QUESTION 8
[tex](4x)^2(b^3)[/tex]
We share the exponent for each factor in the parenthesis.
[tex](4x)^2(b^3)=(4)^2(x^2)(b^3)[/tex]
[tex]\Rightarrow (4a^2)^2(b^3)=16x^{2}b^3[/tex]
ANSWER TO QUESTION 9
[tex](x^2y^4)^5[/tex]
We share the exponent for each factor in the parenthesis.
[tex](x^2y^4)^5=(x^2)^5(y^4)^5[/tex]
[tex](x^2y^4)^5=(x^2)^5(y^4)^5[/tex]
We now use the law,
[tex](a^m)^{n}=a^{(m\times n)}[/tex] to simplify each factor.
[tex](x^2y^4)^5=(x^{2 \times 5})(y^{4 \times 5})[/tex]
[tex](x^2y^4)^5=(x^{10})(y^{20})[/tex]
[tex](x^2y^4)^5=x^{10}y^{20}[/tex]
ANSWER TO QUESTION 10
[tex](2a^3b^2)(b^3)^2[/tex]
Recall that,
[tex](a^m)^{n}=a^{(m\times n)}[/tex]
[tex](2a^3b^2)(b^3)^2=(2a^3b^2)(b^{3 \times 2})[/tex]
[tex](2a^3b^2)(b^3)^2=(2a^3b^2)(b^{6})[/tex]
[tex](2a^3b^2)(b^3)^2=(2a^3b^2)(b^{6})[/tex]
We apply the product law to get,
[tex](2a^3b^2)(b^3)^2=2a^3b^{2+6}[/tex]
[tex](2a^3b^2)(b^3)^2=2a^3b^{8}[/tex]
ANSWER TO QUESTION 11
[tex](-4xy)^3(-2x^2)^3[/tex]
We share the index for each factor to get,
[tex](-4xy)^3(-2x^2)^3=(-4)^3(x^3)(y^3)(-2)^3(x^2)^3[/tex]
We simplify to get,
[tex](-4xy)^3(-2x^2)^3=-64x^3y^3\times -8x^6[/tex]
Applying the product rule gives,
[tex](-4xy)^3(-2x^2)^3=-64\times -8 x^{3+6}y^3[/tex]
[tex](-4xy)^3(-2x^2)^3=512x^{9}y^3[/tex]
ANSWER 12
[tex](-3j^2k^3)^2(2j^2k)^3[/tex]
We split the index for each factor.
[tex](-3j^2k^3)^2(2j^2k)^3=(-3)^2(j^2)^2(k^3)^2(2^3)(j^2)^3(k^3)[/tex]
We simplify to get,
[tex](-3j^2k^3)^2(2j^2k)^3=9\times 8(j^4)(k^6)(j^6)(k^3)[/tex]
[tex](-3j^2k^3)^2(2j^2k)^3=72j^{4+6}k^{6+3}[/tex]
[tex](-3j^2k^3)^2(2j^2k)^3=72j^{10}k^{9}[/tex]
ANSWER 13
[tex](25a^2b)^3(\frac{1}{5}abf)^2[/tex]
We share the index.
[tex](25a^2b)^3(\frac{1}{5}abf)^2=(25^3)(a^2)^3(b^3)(\frac{1}{5})^2a^2b^2f^2)[/tex]
[tex](25a^2b)^3(\frac{1}{5}abf)^2=(25^3)\times (\frac{1}{25}) (a^6)(b^3)a^2b^2f^2[/tex]
[tex](25a^2b)^3(\frac{1}{5}abf)^2=625 (a^{6+2}b^{3+2})[/tex]
[tex](25a^2b)^3(\frac{1}{5}abf)^2=625 (a^{8}b^{5})[/tex]
ANSWER 14.
[tex](2xy)^2(-3x^2)(4y^4)=2^2x^2y^2(-3x^2)(4y^4)[/tex]
[tex](2xy)^2(-3x^2)(4y^4)=4\times -3\times 4 x^{2+2}y^{2+4}[/tex]
[tex](2xy)^2(-3x^2)(4y^4)=-48x^{4}y^{6}[/tex]
SEE ATTACHMENT FOR CONTINUATION
