## Engage NY Eureka Math 8th Grade Module 1 Lesson 5 Answer Key

### Eureka Math Grade 8 Module 1 Lesson 5 Exercise Answer Key

Exercise 1.
Verify the general statement x-b=$$\frac{1}{x^{b}}$$ for x=3 and b=-5.
If b were a positive integer, then we have what the definition states. However, b is a negative integer, specifically
b=-5, so the general statement in this case reads
3-(-5)=$$\frac{1}{3^{-5}}$$.
The right side of this equation is

Since the left side is also 35, both sides are equal.
3-(-5)=$$\frac{1}{3^{-5}}$$=35

Exercise 2.
What is the value of (3×10-2)?
(3×10-2) = 3 × $$\frac{1}{10^{2}}$$ = $$\frac{3}{10^{2}}$$ =0.03

Exercise 3.
What is the value of (3×10-5)?
(3×10-5) = 3×$$\frac{1}{10^{5}}$$ = $$\frac{3}{10^{5}}$$ =0.00003

Exercise 4.
Write the complete expanded form of the decimal 4.728 in exponential notation.
4.728=(4×100)+(7×10-1)+(2×10-2)+(8×10-3)

For Exercises 5–10, write an equivalent expression, in exponential notation, to the one given, and simplify as much as possible.

Exercise 5.
5-3=
$$\frac{1}{5^{3}}$$

Exercise 6.
$$\frac{1}{8^{9}}$$ =
8-9

Exercise 7.
3∙2-4=
3∙$$\frac{1}{2^{4}}$$ =$$\frac{3}{2^{4}}$$

Exercise 8.
Let x be a nonzero number.
x-3=
$$\frac{1}{x^{3}}$$

Exercise 9.
Let x be a nonzero number.
$$\frac{1}{x^{9}}$$ =x-9

Exercise 10.
Let x,y be two nonzero numbers.
xy-4 =
x∙$$\frac{1}{y^{4}}$$ = $$\frac{x}{y^{4}}$$

Exercise 11.
$$\frac{19^{2}}{19^{5}}$$ =
192-5

Exercise 12.
$$\frac{17^{16}}{17^{-3}}$$ =
1716×$$\frac{1}{17^{-3}}$$ =1716×173= 1716+3

Exercise 13.
If we let b=-1 in (11), a be any integer, and y be any nonzero number, what do we get?
(y-1)a=y-a

Exercise 14.
Show directly that ($$\frac{7}{5}$$)-4=$$\frac{7^{-4}}{5^{-4}}$$.
($$\frac{7}{5}$$)-4=(7∙$$\frac{1}{5}$$)-4 By the product formula
=(7∙5-1 )-4 By definition
=7-4∙(5-1 )-4 By (xy)a=xa ya (12)
=7-4∙54 By (xb )a=xab (11)
=7-4∙$$\frac{1}{5-4}$$ By x-b=$$\frac{1}{x^{b}}$$(9)
=$$\frac{7^{-4}}{5^{-4}}$$ By product formula

### Eureka Math Grade 8 Module 1 Lesson 5 Problem Set Answer Key

Question 1.
Compute: 33 ×32 ×31 ×30×3-1 ×3-2=
33 =27
Compute: 52 ×51 0×58 ×50×5-10 ×5-8 =52 =25
Compute for a nonzero number, a: am ×an ×al ×a-n ×a-m ×a-l ×a0=
a0=1

Question 2.
Without using (10), show directly that (17.6-1 )8 = 17.6-8 .
(17.6-1)8 =($$\frac{1}{17.6}$$)8 By definition
= $$\frac{1^{8}}{17.6^{8}}$$ By ($$\frac{x}{y}$$)n = $$\frac{x^{n}}{y^{n}}$$ (5)
= $$\frac{1}{17.6^{8}}$$
= 17.6-8 By definition

Question 3.
Without using (10), show (prove) that for any whole number n and any nonzero number y, (y-1 )n =y-n .
(y-1 )n =($$\frac{1}{y}$$)n By definition
=$$\frac{1^{n}}{y^{n}}$$ By ($$\frac{x}{y}$$)n = $$\frac{x^{n}}{y^{n}}$$(5)
= $$\frac{1}{y^{n}}$$
= y-n By definition

Question 4.
Without using (13), show directly that $$\frac{2.8^{-5}}{2.8^{7}}$$ = 2.8-12 .
$$\frac{2.8^{-5}}{2.8^{7}}$$ = 2.8-5 × $$\frac{1}{2.8^{7}}$$ By the product formula for complex fractions
= $$\frac{1}{2.8^{5}}$$ × $$\frac{1}{2.8^{7}}$$ By definition
= $$\frac{1}{2.8^{5} \times 2.8^{7}}$$ By the product formula for complex fractions
= $$\frac{1}{2.8^{5+7}}$$ By xa∙xb =xa+b (10)
= $$\frac{1}{2.8^{12}}$$
= 2.8-12 By definition

### Eureka Math Grade 8 Module 1 Lesson 5 Exit Ticket Answer Key

Write each expression in a simpler form that is equivalent to the given expression.

Question 1.
76543-4=
$$\frac{1}{76543^{4}}$$

Question 2.
Let f be a nonzero number. f-4=
$$\frac{1}{f^{4}}$$

Question 3.
671×28796-1 =
671×$$\frac{1}{28796}$$=$$\frac{671}{28796}$$
a∙$$\frac{1}{b}$$=$$\frac{a}{b}$$
Let g be a nonzero number. $$\frac{1}{g^{-1}}$$ =