Exercise 22 Page 50

We are asked to find which of the given expressions have values less than 1 when x=4. & (3/x^2)^0 & x^0/3^2 & 1/6^(- x) & 1/x^(-3) & 3x^(-4) To do that we will evaluate them one at a time, starting with ( 3x^2)^0.

(3/x^2)^0

To evaluate the first expression, let's start with substituting x= 4. (3/x^2)^0=(3/4^2)^0 Now, let's recall the Zero Exponent Property.

Zero Exponent Property

Any non-zero real number raised to the power of 0 is equal to 1. a^0=1

Notice that the expression ( 34^2)^0 is a non-zero real number raised to the power of 0. Therefore, according to the Zero Exponent Property, the expression must be equal to 1. (3/4^2)^0=1 Since 1 is not less than 1, we can note that the expression ( 3x^2)^0 does not have a value less than 1 when x=4. &* (3/x^2)^0 & x^0/3^2 & 1/6^(- x) & 1/x^(-3) & 3x^(-4)

x^0/3^2

Next, we will evaluate x^0/3^2 when x=4. Similar to before, we can first substitute x=4 into the expression.

x^0/3^2=4^0/3^2 Now we can use the Zero Exponent Property again. 4^0/3^2=1/3^2 Finally, we can simplify the denominator. 1/3^2=1/9 We found that when x=4 the expression x^03^2 is equal to 19, which is less than 1. &* (3/x^2)^0 &✓ x^0/3^2 & 1/6^(- x) & 1/x^(-3) & 3x^(-4)

1/6^(- x)

Let's evaluate 16^(- x). As before, we will first substitute x= 4. 1/6^(- x)=1/6^(- 4) Next, let's recall the Negative Exponent Property.

Negative Exponent Property

If a is a non-zero real number, then the following equation is true. a^(- n)=1/a^n

We can apply this property to the denominator of our expression. 1/6^(- 4)=1/16^4 Dividing 1 by the fraction 16^4 is the same as multiplying 1 by the reciprocal of the fraction, which is 6^41. 1/16^4=1* 6^4/1 Let's simplify this a bit more.

1* 6^4/1

DivByOne

a/1=a

1* 6^4

MultByOne

a * 1=a

6^4

Rewrite

Rewrite 6^4 as 6*6*6*6

6*6*6*6

Multiply

Multiply

1296

We found that when x=4 the expression 1x^(- x) is equal to 1296, which is not less than one. Let's mark this result in our list. &* (3/x^2)^0 &✓ x^0/3^2 &* 1/6^(- x) & 1/x^(-3) & 3x^(-4)

1/x^(- 3)

Now, we will calculate 1x^(- 3) when x=4. Let's substitute! 1/x^(- 3)=1/4^(- 3) Again, we will use the Negative Exponent Property. 1/4^(- 3)=1/14^3 We can calculate this quotient by multiplying 1 by the reciprocal of the denominator. 1/14^3=1* 4^3/1 Let's continue simplifying the expression.

1* 4^3/1

DivByOne

a/1=a

1* 4^3

MultByOne

a * 1=a

4^3

Rewrite

Rewrite 4^3 as 4*4*4

4*4*4

Multiply

Multiply

64

When x=4 the expression 1x^(- 3) is equal to 64, which is not less than one. We can mark this result in our list. &* (3/x^2)^0 &✓ x^0/3^2 &* 1/6^(- x) &* 1/x^(-3) & 3x^(-4)

3x^(-4)

Finally, we will check if 3x^(-4) is less than 1 when x=4. Just like before, the first step is to substitute x= 4. 3x^(-4)=3( 4)^(-4) Next, let's use the Negative Exponent Property. 3(4)^(-4)=3* 1/4^4 Let's try to simplify this.

3*1/4^4

MoveLeftFacToNum

a*b/c= a* b/c

3/4^4

Rewrite

Rewrite 4^4 as 4*4*4*4

3/4*4*4*4

Multiply

Multiply

3/256

We found that when x=4 the expression 3x^(- 4) is equal to 3256, which is less than 1. Let's mark this in our list. &* (3/x^2)^0 &✓ x^0/3^2 &* 1/6^(- x) &* 1/x^(-3) &✓ 3x^(-4) We found that only the expressions x^03^2 and 3x^(-4) are less than 1 when x=4.