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

Variations Integral Exponents Radicals

Variations

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

Graph

Direct Square Variation

Graph

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

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.

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

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.

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

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.

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!!!

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

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 19.57 meters. How far will it fall in 5 seconds? d = k 𝒕 𝟐 19.57 = k (𝟐) 𝟐 19.57 = k4 4.8925 = k d = (4.8925) (𝟓) 𝟐 d = 122.31 𝒚 𝟏 𝒙 𝟏 𝟐 = 𝒚 𝟐 𝒙 𝟐 𝟐 𝟏𝟗.𝟓𝟕 𝟐 𝟐 = 𝒚 𝟐 𝟓 𝟐 𝟒𝒚 𝟐 = 489.25 𝒚 𝟐 = 122.31

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

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

Integral Exponent

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

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

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

Radicals

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 .

Example:

Example:

Example:

Example:

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 + 3 𝟑𝟐𝒂 + 4 𝟏𝟐𝒃 + 2 𝟐𝟕𝒃 Simplify 𝟐𝟎 + 𝟒𝟓 + 𝟖𝟎 - 10 𝟏𝟐𝟓 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 +14 𝟑𝒃 -41 𝟓 24 𝟑 𝟒 𝟏𝟖 2 𝟑 𝒎 𝟐 𝟓𝒚 𝟐 𝒙 𝟐 𝟐 𝟑𝒙