Chapter 1.2, Problem 43E

(a)

To determine

To simplify: given expression $\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4$ .

Answer to Problem 43E

  $\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4=405x^{10}{y}^{23}$

Explanation of Solution

Given information:

Given expression

  $\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4$

Calculation:

Consider the given expression.

  $\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4$

Now. Simplify using law of exponents ${\left(ab\right)}^n=a^n{b}^n$ and ${\left(a^m\right)}^n=a^{mn}$ .

  $\begin{array}{c}\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4=\left(5x^2{y}^3\right)3^4{\left(x^2\right)}^4{\left(y^5\right)}^4\\ =\left(5x^2{y}^3\right)3^4\left(x^8\right)\left(y^{20}\right)\end{array}$

To simplify this, rewrite the above expression as a single product.

  $\begin{array}{c}\left(5x^2{y}^3\right)3^4\left(x^8\right)\left(y^{20}\right)=5\cdot 3^4\cdot x^2\cdot x^8\cdot y^3\cdot y^{20}\\ =405\cdot x^2\cdot x^8\cdot y^3\cdot y^{20}\end{array}$

Now, use the first law of exponents, which states

  $a^m{a}^n=a^{m+n}$

And combine like terms

  $\begin{array}{c}405\cdot x^2\cdot x^8\cdot y^3\cdot y^{20}=405\cdot x^{2+8}\cdot y^{20+3}\\ =405\cdot x^{10}\cdot y^{23}\end{array}$

Hence,

  $\left(5x^2{y}^3\right){\left(3x^2{y}^5\right)}^4=405x^{10}{y}^{23}$

(b)

To determine

To evaluate: the given expression ${\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3$ .

Answer to Problem 43E

  ${\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3=500a^{12}{b}^{19}$

Explanation of Solution

Given information:

Given expression

  ${\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3$

Calculation:

Consider the given expression.

  ${\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3$

Now. Simplify using law of exponents ${\left(ab\right)}^n=a^n{b}^n$ and ${\left(a^m\right)}^n=a^{mn}$ .

  $\begin{array}{c}{\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3=\left\{2^2{\left(a^3\right)}^2{\left(b^2\right)}^2\right\}\left\{5^3{\left(a^2\right)}^3{\left(b^5\right)}^3\right\}\\ =\left(4a^6{b}^4\right)\left(5^3{a}^6{b}^{15}\right)\end{array}$

To simplify this, first rewrite the above expression as a single product.

  $\begin{array}{c}\left(4a^6{b}^4\right)\left(5^3{a}^6{b}^{15}\right)=4\cdot 5^3\cdot a^6\cdot a^6\cdot b^4\cdot b^{15}\\ =500\cdot a^6\cdot a^6\cdot b^4\cdot b^{15}\end{array}$

Now, use the first law of exponents, which states

  $a^m{a}^n=a^{m+n}$

And combine like terms

  $\begin{array}{c}500\cdot a^6\cdot a^6\cdot b^4\cdot b^{15}=500\cdot a^{6+6}\cdot b^{4+15}\\ =500\cdot a^{12}\cdot b^{19}\end{array}$

Hence,

  ${\left(2a^3{b}^2\right)}^2{\left(5a^2{b}^5\right)}^3=500a^{12}{b}^{19}$