Chapter ISG, Problem 7.1STP

a.

To determine

To find the missing exponent

Answer to Problem 7.1STP

Missing exponents are $x=7,y=2$

Explanation of Solution

Given:

  $\frac{{(a^4{b}^{\_})}^3}{a^{5}b}=a^{\_}{b}^5$

Calculation:

Let the missing exponent of a be x and missing exponent of b be y

According to given expression

  $\frac{{(a^4{b}^{\_})}^3}{a^{5}b}=a^{\_}{b}^5$

Solving the equation,

  $\frac{a^{4\times 3}{b}^{\_\times 3}}{a^{5}b}=a^{\_}{b}^5$

So,

  $\frac{a^{12}{b}^{\_\times 3}}{a^{5}b}=a^{\_}{b}^5$

Dividing,

  $a^{12-5}{b}^{(\_\times 3)-1}=a^{\_}{b}^5$

Comparing both sides as bases are same,

  $12-5=x$ and $3y-1=5$

Solving,

  $x=7,3y=6$

So,

  $x=7,y=2$

Hence, missing exponents are $x=7,y=2$

b.

To determine

To find the missing exponents

Answer to Problem 7.1STP

  $a=6,b=2$ and $c=3$

Explanation of Solution

Given:

  $x^{-2}{(y^{2}z)}^3=\frac{x^4{y}^{\_}{z}^{\_}}{x^{\_}{y}^{-1}}$

Calculation:

Let the missing values of x, y, z be a, b, c.

The given expression is,

  $x^{-2}{(y^{2}z)}^3=\frac{x^4{y}^b{z}^c}{x^a{y}^{-1}}$

Solving the equation,

  $x^{-2}{y}^{2(3)}{z}^3=\frac{x^4{y}^b{z}^c}{x^a{y}^{-1}}$

Dividing the expression,

  $x^{-2}{y}^6{z}^3=x^{4-a}{y}^{b-(-1)}{z}^c$

As the bases are equal, compare the power.

  $-2=4-a,3=b+1$ and $3=c$

Solving,

  $a=6,b=2$ and $c=3$

Hence, $a=6,b=2$ and $c=3$ are the missing values.