Eureka Math Grade 8 Module 1 Lesson 5 Answer Key

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.
Answer:
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
Eureka Math Grade 8 Module 1 Lesson 5 Exercise Answer Key 1
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)?
Answer:
(3×10-2) = 3 × \(\frac{1}{10^{2}}\) = \(\frac{3}{10^{2}}\) =0.03

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

Exercise 6.
\(\frac{1}{8^{9}}\) =
Answer:
8-9

Exercise 7.
3∙2-4=
Answer:
3∙\(\frac{1}{2^{4}}\) =\(\frac{3}{2^{4}}\)

Exercise 8.
Let x be a nonzero number.
x-3=
Answer:
\(\frac{1}{x^{3}}\)

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 =
Answer:
x∙\(\frac{1}{y^{4}}\) = \(\frac{x}{y^{4}}\)

Exercise 11.
\(\frac{19^{2}}{19^{5}}\) =
Answer:
192-5

Exercise 12.
\(\frac{17^{16}}{17^{-3}}\) =
Answer:
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?
Answer:
(y-1)a=y-a

Exercise 14.
Show directly that (\(\frac{7}{5}\))-4=\(\frac{7^{-4}}{5^{-4}}\).
Answer:
(\(\frac{7}{5}\))-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=
Answer:
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=
Answer:
a0=1

Question 2.
Without using (10), show directly that (17.6-1 )8 = 17.6-8 .
Answer:
(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 .
Answer:
(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 .
Answer:
\(\frac{2.8^{-5}}{2.8^{7}}\) = 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=
Answer:
\(\frac{1}{76543^{4}}\)

Question 2.
Let f be a nonzero number. f-4=
Answer:
\(\frac{1}{f^{4}}\)

Question 3.
671×28796-1 =
Answer:
671×\(\frac{1}{28796}\)=\(\frac{671}{28796}\)

Question 4.
Let a, b be numbers (b≠0). ab-1 =
Answer:
a∙\(\frac{1}{b}\)=\(\frac{a}{b}\)

Question 5.
Let g be a nonzero number. \(\frac{1}{g^{-1}}\) =
Answer:
g

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