Exercise 5 Page &nbsp; 237

Split into factors

4^(1/2 * 3)

a^(m* n)=(a^m)^n

(4^(1/2))^3

a^(1/2)=sqrt(a)

(sqrt(4))^3

Having rewritten the expression as a power involving a radical, it is now easier to calculate. The square root of 4 is the number you have to multiply by itself to obtain 4, which is 2.

(sqrt(4))^3

Write as a power

(sqrt(2^2))^3

SqrtPowToNumber

sqrt(a^2)=a

2^3

Next, we want to calculate 2^3, which is 2 multiplied by itself 3 times.

2^3

a^3=a* a* a

2* 2* 2

Add parentheses

(2* 2)* 2

4* 2

b To evaluate the expression without a calculator, we should first rewrite it as a power involving a radical.

32^(4/5)

Split into factors

32^(1/5 * 4)

a^(m* n)=(a^m)^n

(32^(1/5))^4

PowToRootSL

a^(1/n)=sqrt(a)

(sqrt(32))^4

Having rewritten the expression as a power involving a radical, it is now easier to calculate. The fifth root of 32 is the number you have to multiply by itself five times to obtain 32, which is 2.

(sqrt(32))^4

Write as a power

(sqrt(2^5))^4

RootPowToNumber

sqrt(a^n)=a

2^4

Next we want to calculate 2^4, which is 2 multiplied by itself 4 times.

2^4

Split into factors

2* 2* 2* 2

Add parentheses

(2* 2)* (2* 2)

4* 4

c Like in Parts A and B, we should first rewrite the expression as a power involving a radical.

625^(3/4)

Split into factors

625^(1/4 * 3)

a^(m* n)=(a^m)^n

(625^(1/4))^3

PowToRootSL

a^(1/n)=sqrt(a)

(sqrt(625))^3

Having rewritten the expression as a power involving a radical, it is now easier to calculate. The fourth root of 625 is the number you have to multiply by itself four times to obtain 625, which is 5.

(sqrt(625))^3

Write as a power

(sqrt(5^4))^3

RootPowToNumber

sqrt(a^n)=a

5^3

Next we want to calculate 5^3, which is 5 multiplied by itself 3 times.

5^3

a^3=a* a* a

5* 5* 5

Add parentheses

(5* 5)* 5

25* 5

125

d Like in Parts A through C, we should first rewrite the expression as a power involving a radical.

49^(3/2)

Split into factors

49^(1/2 * 3)

a^(m* n)=(a^m)^n

(49^(1/2))^3

a^(1/2)=sqrt(a)

(sqrt(49))^3

Having rewritten the expression as a power involving a radical, it is now easier to calculate. The square root of 49 is the number you have to multiply by itself to obtain 49, which is 7.

(sqrt(49))^3

Write as a power

(sqrt(7^2))^3

SqrtPowToNumber

sqrt(a^2)=a

7^3

Next we want to calculate 7^3, which is 7 multiplied by itself 3 times.

7^3

a^3=a* a* a

7* 7* 7

Add parentheses

(7* 7)* 7

49* 7

Note that this expression is not very easy to calculate. However, if we rewrite 49 as 50-1 and distribute 7, we get easier numbers to work with.

49* 7

WriteDiff

Write as a difference

(50-1)7

Distr

Distribute 7

3500-7

SubTerm

Subtract term

3493

e Like in Parts A through D, we should first rewrite the expression as a power involving a radical.

125^(4/3)

Split into factors

125^(1/3 * 4)

a^(m* n)=(a^m)^n

(125^(1/3))^4

PowToRootSL

a^(1/n)=sqrt(a)

(sqrt(125))^4

Having rewritten the expression as a power involving a radical, it is now easier to calculate. The third root of 125 is the number you have to multiply by itself three times to obtain 125, which is 5.

(sqrt(125))^4

Write as a power

(sqrt(5^3))^4

RootPowToNumber

sqrt(a^n)=a

5^4

Next we want to calculate 5^4, which is 5 multiplied by itself 4 times.

5^4

Split into factors

5* 5* 5* 5

Add parentheses

(5* 5)* (5* 5)