1. REVIEW
FOR
SECOND QUARTER
Prepared by:
Mr Allan P. Limin

2. Variations
Integral Exponents
Radicals

3. Variations

4. Direct Variation
y = kx or k = 𝒚 𝒙
Direct Square Variation
y = 𝒌𝒙 𝟐 or k = 𝒚 𝒙 𝟐
Inverse Variation
y = 𝒌 𝒙 or k = xy
Joint Variation
y = kxz or k = 𝒚 𝒙𝒛
Combined Variation
y = 𝒌𝒙 𝒛 or k = 𝒚𝒛 𝒙

6. Graph

7. Direct Square Variation

9. Graph

10. y = kxz
20 = k (4)(3)
20 = k 12
k = 𝟓 𝟑
y = kxz
y = ( 𝟓 𝟑 )(2)(3)
y = 10

12. Problem:
A. Write each in equation form.
The volume V of a cylinder varies directly as its height h.
The area A of a square varies directly as the square of its
side s.
The length l of a rectangular field varies inversely as its
width w.
The volume of cylinder V varies jointly as its height h
and the square of the radius r.
The electrical resistance R of a wire varies directly as its
length l and inversely as the square of its diameter d.

13. V = kh
Answers
2. A = k 𝒔 𝟐
3. l = 𝒌 𝒘
4. V = kh 𝒓 𝟐
5. R = 𝒌𝒍 𝒅 𝟐 .

14. B. Find k and express the equation of variation.
y varies directly as x and y=30 when x=8.
x varies directly as the square of y, and x=6, when y=8.
c varies jointly as a and b, and c=45, a=15 and b=14.
x varies directly as y and inversely as z, when x=15, y=20 and z=40.

15. 1. y = kx
30 = k8
𝟏𝟓 𝟒 = k
y = 𝟏𝟓 𝟒 x
2. x = k 𝒚 𝟐
6 = k (𝟖) 𝟐
6 = k64
𝟑 𝟑𝟐 = k
x = 𝟑 𝟑𝟐 𝒚 𝟐
3. c = kab
45 = k(15)(14)
45 = k 210
𝟑 𝟏𝟒 = k
c = 𝟑 𝟏𝟒 ab
4. x = 𝒌𝒚 𝒛
15 = 𝒌𝟐𝟎 𝟒𝟎
30 = k
x = 𝟑𝟎𝒚 𝒛

16. C. Solve for the indicated variable in each of the ff.
If y varies directly as x, and y = -18 when x = 9, find y when x = 7.
If y varies directly as the square of x, and y=36 when x=3, find y when x=5.
z varies jointly as x and y, and z=60 when x=5, y=6, find z when x=7 and y=6.
If r varies directly as s and inversely as the square of u, and r=2 when s=18 and u=2, find r when u=3 and s =27.

17. Answers!!! 2. y = k 𝒙 𝟐 36 =k (𝟑) 𝟐 36 =k9 4 = k y = (4) (𝟓) 𝟐 y = 100
1. y = kx
-18 = k9
-2 = k
y = (-2)(7)
y = -14
3. z = kxy
60 = k(5)(6)
60 = k 30
2 = k
z = (2)(7)(6)
z = 84
4. r = 𝒌𝒔 𝒖 𝟐
2 = 𝒌𝟏𝟖 (𝟐) 𝟐
𝟒 𝟗 = k
r = 𝟒 𝟗 (𝟐𝟕) (𝟑) 𝟐
r = 𝟏𝟐 𝟗
r = 𝟒 𝟑
Answers!!!

18. Worded Problems
Candies are sold at 50 centavos each. How much
will a bag of 420 candies cost?
y = kx
.50 = k1
.50 = k
y = (.50)(420)
y = ₱210.00
𝒚 𝟏 𝒙 𝟏 = 𝒚 𝟐 𝒙 𝟐
.𝟓𝟎 𝟏 = 𝒚 𝟐 𝟒𝟐𝟎
𝒚 𝟐 = ₱210.00

19. 2. When a body falls from rest, its distance from the
starting point is directly proportional to the square
of the time during which it is falling. In 2 seconds, a
body falls through meters. How far will it fall
in 5 seconds?
d = k 𝒕 𝟐
19.57 = k (𝟐) 𝟐
19.57 = k4
= k
d = (4.8925) (𝟓) 𝟐
d =
𝒚 𝟏 𝒙 𝟏 𝟐 = 𝒚 𝟐 𝒙 𝟐 𝟐
𝟏𝟗.𝟓𝟕 𝟐 𝟐 = 𝒚 𝟐 𝟓 𝟐
𝟒𝒚 𝟐 =
𝒚 𝟐 =

20. 3. The mass of a rectangular sheet of wood varies jointly as the length and the width. When the length is 20 cm and the width is 10 cm, the mass is 200 g. Find the mass when the length is 15 cm and the width is 10 cm.
m = klw
200 = k(20)(10)
200 = k200
1 = k
m = (1)(15)(10)
m = 150 grams
𝒎 𝟏 𝒍 𝟏 𝒘 𝟏 = 𝒎 𝟐 𝒍 𝟐 𝒘 𝟐
𝟐𝟎𝟎 (𝟐𝟎)(𝟏𝟎) = 𝒎 𝟐 (𝟏𝟓)(𝟏𝟎)
200 𝒎 𝟐 = 30,000
𝒎 𝟐 = 150 grams

21. 4. The current I varies directly as the electromotive force E
And inversely as the resistance R. If in a system a current of
20 amperes flows through a resistance of 20 ohms with an
Electromotive force of 100 volts, find the current that 150 volts will send through the system.
I = 𝒌𝑬 𝑹
20 = 𝒌𝟏𝟎𝟎 𝟐𝟎
400 = k100
4 = k
I = 𝒌𝑬 𝑹
I = (𝟒)𝟏𝟓𝟎 𝟐𝟎
I = 30
𝑰 𝟏 𝑹 𝟏 𝑬 𝟏 = 𝑰 𝟐 𝑹 𝟐 𝑬 𝟐
(𝟐𝟎)(𝟐𝟎) 𝟏𝟎𝟎 = 𝑰 𝟐 𝟐𝟎 𝟏𝟓𝟎
60,000 = 𝑰 𝟐 2,000
30 = 𝑰 𝟐

22. Integral Exponent

23. Laws of Exponents
1. Product of Powers: 𝒙 𝒎 ∗ 𝒙 𝒏 = 𝒙 𝒎+𝒏
2. Power of a Power: (𝒙 𝒎 ) 𝒏 = 𝒙 𝒎𝒏
3. Power of a Product: (𝒂𝒃) 𝒎 = 𝒂 𝒎 𝑏 𝒎
4. Power of a Monomial: ( 𝒂 𝒎 𝒃 𝒏 ) 𝒑 = 𝒂 𝒎𝒑 𝑏 𝒏𝒑
5. Quotient of Powers: 𝒂 𝒎 𝒂 𝒏 = 𝒂 𝒎−𝒏

24. 6. Power of a Fraction: ( 𝒂 𝒃 ) 𝒎 = 𝒂 𝒎 𝒃 𝒎
7. Zero Exponent: 𝒂 𝟎 =𝟏
8. Negative Exponent: 𝒂 −𝒏 = 𝟏 𝒂 𝒏
9. Rational Exponent: (𝒃 𝟏 𝒏 ) 𝒎 = 𝒃 𝒎 𝒏 or 𝒃 −𝒎 𝒏 = 𝟏 𝒃 𝒎 𝒏

25. Evaluate: 𝒃 𝟓 ∗ 𝒃 𝟑 (𝒎 𝟑 ) 𝟓 (𝟐𝒂) 𝟑 ( 𝒂 𝟒 𝒃 𝟑 ) 𝟑 𝟏𝟎 𝟏𝟎 𝟏𝟎 𝟖 ( 𝒂 𝒃 ) 𝟔
(𝒎 𝟑 ) 𝟓
(𝟐𝒂) 𝟑
( 𝒂 𝟒 𝒃 𝟑 ) 𝟑
𝟏𝟎 𝟏𝟎 𝟏𝟎 𝟖
( 𝒂 𝒃 ) 𝟔
𝟖𝒙 𝟐 𝒚 𝟎
𝟗 −𝟐
𝟔𝒙 −𝟑 𝒚 𝟑𝒙 𝟐 𝒚 −𝟒
1. 𝒃 𝟖
2. 𝒎 𝟏𝟓
3. 8 𝒂 𝟑
4. 𝒂 𝟏𝟐 𝒃 𝟗
5. 100
6. 𝒂 𝟔 𝒃 𝟔
7. 𝟖𝒙 𝟐
8. 𝟏 𝟖𝟏
9. 𝟐𝒚 𝟓 𝒙 𝟓

26. Radicals

27. Definition:
If a is a nonnegative real number, the nonnegative number b such that 𝒃 𝟐 = a is the principal square root of a and
Is denoted by b = 𝒂 .

29. Example:

30. Example:

31. Example:

32. Example:

35. Find the product of ( 𝟒 𝟐 )( 𝟑 ) Rationalize the denominator of 𝟐𝒎 𝟑 𝒎
Evaluate:
𝒂 𝟏 𝟐
The equivalent of 𝒂
Simplify 𝟐𝟕𝒙
Evaluate 𝟒𝟗𝒚 𝟏𝟎
Simplify ( 𝟑 𝟔𝟒 ) −𝟐
Simplify 2 𝟏𝟖𝒂 𝟑𝟐𝒂 𝟏𝟐𝒃 𝟐𝟕𝒃
Simplify 𝟐𝟎 + 𝟒𝟓 + 𝟖𝟎 𝟏𝟐𝟓
Multiply (4 𝟔 )(3 𝟐 )
Find the product of ( 𝟒 𝟐 )( 𝟑 )
Rationalize the denominator of 𝟐𝒎 𝟑 𝒎
Divide: 𝟓𝟎𝒙 −𝟒 𝒚 𝟕 𝟑𝒙𝒚 𝟐
3 𝟑𝒙
𝟕𝒚 𝟓
1/16
18 𝟐𝒂 𝟑𝒃
-41 𝟓
24 𝟑
𝟒 𝟏𝟖
2 𝟑 𝒎 𝟐
𝟓𝒚 𝟐 𝒙 𝟐 𝟐 𝟑𝒙