Chapter ISG, Problem 7.1.3P

a.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Not equivalent

Explanation of Solution

Given:

  $(2^7)(2^7)(2^7)(2^7)$ and $8^7$

Formula used:

a^maⁿ = a^m + n for a ≠ 0 or m , n ≥ 0;

Calculation:

Left hand side,

Using identity and substituting value,

  = ${(2)}^{7+7+7+7}$

Solving,

  = ${(2)}^{28}$

Right hand side,

  = $8^7$

Solving,

  = ${({(2)}^3)}_{}^7$

So, using identity ( a^m ) n = a^mn for a ≠ 0 or m ≥ 0;

  =(2)21

Hence, left hand side $\ne$ right hand side.

b.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Equivalent

Explanation of Solution

Given:

  ${(-j)}^9{(-j)}^9{(-j)}^9$ and ${(-j)}^{27}$

Formula used:

a^maⁿ = a^m + n for a ≠ 0 or m , n ≥ 0;

Calculation:

Left hand side,

Using identity and substituting value,

  = ${(-j)}^{9+9+9}$

Solving,

  = ${(-j)}^{27}$

Right hand side,

  = $-j^{27}$

Hence, left hand side= right hand side.

c.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Equivalent

Explanation of Solution

Given:

  $(5p^{3}q)(4p^5{q}^9)$ and $20p^8{q}^{10}$

Formula used:

a^maⁿ = a^m + n for a ≠ 0 or m , n ≥ 0;

Calculation:

Left hand side,

Putting all like terms together,

  $5\times 4(p^3)(p^5)(q)(q^9)$

Using identity and substituting value,

  = $20(p^{3+5})(q^{1+9})$

Solving,

  = $20p^8{q}^{10}$

Right hand side,

  = $20p^8{q}^{10}$

Hence, left hand side=right hand side.

d.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Not equivalent

Explanation of Solution

Given:

  ${(6w^5)}^2$ and $36w^{25}$

Formula used:

a^maⁿ = a^m + n for a ≠ 0 or m , n ≥ 0;

( a^m ) n = a^mn for a ≠ 0 or m ≥ 0;

Calculation:

Left hand side,

Using identity and substituting value,

  = ${(6w^5)}^2$

Solving,

  = $(6^2){(w^5)}^2$

So,

  = $36w^{5\times 2}$

And,

  = $36w^{10}$

Right hand side,

  = $36w^{25}$

Hence, left hand side $\ne$ right hand side.

e.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Equivalent

Explanation of Solution

Given:

  ${(x^{10})}^3$ and ${(x^3)}^{10}$

Formula used:

( a^m ) n = a^mn for a ≠ 0 or m ≥ 0;

Calculation:

Left hand side,

Using identity and substituting value,

  = ${(x^{10})}^3$

Solving,

  = $x^{10\times 3}$

So,

  = $x^{30}$

Right hand side,

Using identity and substituting value,

  = ${(x^3)}^{10}$

Solving,

  = $x^{3\times 10}$

So,

  = $x^{30}$

Hence, left hand side=right hand side.

f.

To determine

To find which pair of expressions are equivalent.

Answer to Problem 7.1.3P

Equivalent

Explanation of Solution

Given:

  ${[{\left\{2n^2\right\}}^3]}^2$ and ${(4n_{}^4)}^3$

Formula used:

( a^m ) n = a^mn for a ≠ 0 or m ≥ 0;

Calculation:

Left hand side,

Using identity and substituting value,

  = ${[{\left\{2n^2\right\}}^3]}^2$

Solving,

  = ${[{(2)}^3{n}^{2(3)}]}^2$

So,

  = ${(2)}^{3(2)}{(n)}^6{}^{(2)}$

And,

  = ${(2)}^2{(n)}^{12}$

Right hand side,

Using identity and substituting value,

  = ${(4n_{}^4)}^3$

Solving,

  = $4^3{n}^{4(3)}$

So,

  = ${(2^2)}^3{n}^{12}$

  = $2^6{n}^{12}$

Hence, left hand side=right hand side.