text: EXPONENTS AND SQUARE ROOTS WORKSHEET
  
  About "Exponents and Square Roots Worksheet"
  
  Exponents and Square Roots Worksheet :
  
  Worksheet given in this section is much useful to the students who would like to practice problems on exponents and square roots.
  
  Before look at the worksheet, if you would like to learn the stuff exponents and square roots in detail,
  
  Exponents and Square Roots Worksheet - Problems
  
  Problem 1 :
  
  Simplify the following :
  
  (-4)5 ÷ (-4)8
  
  Problem 2 :
  
  If a-1/2  =  5, then find the value of a.
  
  Problem 3 :
  
  If x2  =  y3 and x3z  =  y9, then find the value of z.
  
  Problem 4 :
  
  Solve for k :
  
  3x2 ⋅ 2x3  =  6xk
  
  Problem 5 :
  
  Rationalize the denominator :
  
  / (3 + √2)
  
  Problem 6 :
  
  Rationalize the denominator :
  
  (1 - √5) / (3 + √5)
  
  Problem 7 :
  
  Find the value of :
  
  √2.56
  
  Problem 8 :
  
  Simplify the following square root expression :
  
  (√17)(√51)
  
  Problem 9 :
  
  If x2y3  =  10 and x3y2  =  8, then find the value of x5y5.
  
  Problem 10 :
  
  If (√9)-7 ⋅ (√3)-4  =  3k, then solve for k.
  
  Exponents and Square Roots Worksheet - Solutions
  
  Problem 1 :
  
  Simplify the following :
  
  (-4)5 ÷ (-4)8
  
  Solution :
  
  (-4)5 ÷ (-4) =  (-4)5-8
  
  (-4)5 ÷ (-4) =  (-4)-3
  
  (-4)5 ÷ (-4) =  1 / (-4)3
  
  (-4)5 ÷ (-4) =  1 / (-64)
  
  (-4)5 ÷ (-4) =  - 1 / 64
  
  Problem 2 :
  
  If a-1/2  =  5, then find the value of a.
  
  Solution :
  
  a-1/2  =  5
  
  a  =  5-2/1
  
  a  =  5-2
  
  a  =  1/52
  
  a  =  1/25
  
  Problem 3 :
  
  If x2  =  y3 and x3z  =  y9, then find the value of z.
  
  Solution :
  
  x3z  =  y9
  
  x3z  =  y⋅ 3
  
  x3z  =  (y3)3
  
  Substitute xfor y3.
  
  x3z  =  (x2)3
  
  x3z  =  x6
  
  3z  =  6
  
  Divide each side by 3.
  
  z  =  2
  
  So, the value of z is 2.
  
  Problem 4 :
  
  Solve for k :
  
  3x2 ⋅ 2x3  =  6xk
  
  Solution :
  
  3x2 ⋅ 2x3  =  6xk
  
  (3 ⋅ 2)(x⋅ x3)  =  6xk
  
  (6)(x2+3)  =  6xk
  
  6x5  =  6xk
  
  Divide each side by 6.
  
  x5  =  xk
  
  Using laws of exponents, we have
  
  5  =  k
  
  So, the value of k is 5.
  
  Problem 5 :
  
  Rationalize the denominator :
  
  / (3 + √2)
  
  Solution :
  
  To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √2), that is by (3 - √2).
  
  1 / (3 + √2)  =  [1 ⋅ (3-√2)] / [(3+√2)  (3-√2)]
  
  1 / (3 + √2)  =  (3-√2) / [(3+√2)  (3-√2)]
  
  Using the algebraic identity a2 - b2  =  (a + b)(a - b), simplify the denominator on the right side.
  
  1 / (3 + √2)  =  (3-√2) / [32 - (√2)2]
  
  1 / (3 + √2)  =  (3-√2) / (9 - 2)
  
  1 / (3 + √2)  =  (3 - √2) / 7
  
  Problem 6 :
  
  Rationalize the denominator :
  
  (1 - √5) / (3 + √5)
  
  Solution :
  
  To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √5), that is by (3 - √5).
  
  (1 - √5) / (3 + √5)  =  [(1-√5) ⋅ (3-√5)] / [(3+√5)  (3-√5)]
  
  Simplify.
  
  (1 - √5) / (3 + √5)  =  [3 √5 - 3√5 + 5] / [32 - (√5)2]
  
  (1 - √5) / (3 + √5)  =  (8 - 4√5) / (9 - 5)
  
  (1 - √5) / (3 + √5)  =  4(2 - √5) / 4
  
  (1 - √5) / (3 + √5)  =  2 - √5
  
  Problem 7 :
  
  Find the value of :
  
  √2.56
  
  Solution :
  
  To get rid of the decimal, multiply and divide 2.56 inside the radical by 100.
  
  √2.56  =  √(2.56  100 / 100)
  
  √2.56  =  √(256 / 100)
  
  √2.56   √(16  16) / (10 ⋅ 10)
  
  √2.56  =  16 / 10
  
  √2.56  =  1.6
  
  Problem 8 :
  
  Simplify the following square root expression :
  
  (√17)(√51)
  
  Solution :
  
  Decompose 17 and 51 into prime factors.
  
  Because 17 is a prime number, it can't be decomposed anymore. So, √17 has to be kept as it is.
  
  √51  =  √(3 ⋅ 17)  =  √3 ⋅ √17
  
  So, we have
  
  (√17)(√51)  =  (√17)(√3 ⋅ √17)
  
  (√17)(√51)  =  (√17 ⋅ √17)√3
  
  (√17)(√51)  =  17√3
  
  Problem 9 :
  
  If x2y3  =  10 and x3y2  =  8, then find the value of x5y5.
  
  Solution :
  
  x2y3  =  10 -----(1)
  
  x3y2  =  8 -----(2)
  
  Multiply (1) and (2) :
  
  (1) ⋅ (2) -----> (x2y3) ⋅ (x3y2)  =  10 ⋅ 8
  
  x5y5  =  80
  
  So, the value x5yis 80.
  
  Problem 10 :
  
  If (√9)-7 ⋅ (√3)-4  =  3k, then solve for k.
  
  Solution :
  
  (91/2)-7 ⋅ (31/2)-4  =  3k
  
  (9)-7/2 ⋅ (3)-4/2  =  3k
  
  (32)-7/2 ⋅ 3-2  =  3k
  
  3⋅ (-7/2) ⋅ 3-2  =  3k
  
  3-7 ⋅ 3-2  =  3k
  
  3-7 - 2  =  3k
  
  3-9  =  3k
  
  k  =  -9
  
  So, the value of k is -9.
  
  After having gone through the stuff given above, we hope that the students would have understood "Exponents and Square Roots Worksheet".
  
  Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
  
  You can also visit our following web pages on different stuff in math.
  
  WORD PROBLEMS
  
  Word problems on simple equations
  
  Word problems on linear equations
  
  Algebra word problems
  
  Word problems on trains
  
  Area and perimeter word problems
  
  Word problems on direct variation and inverse variation
  
  Word problems on unit price
  
  Word problems on unit rate
  
  Word problems on comparing rates
  
  Converting customary units word problems
  
  Converting metric units word problems
  
  Word problems on simple interest
  
  Word problems on compound interest
  
  Word problems on types of angles
  
  Complementary and supplementary angles word problems
  
  Double facts word problems
  
  Trigonometry word problems
  
  Percentage word problems
  
  Profit and loss word problems
  
  Markup and markdown word problems
  
  Decimal word problems
  
  Word problems on fractions
  
  Word problems on mixed fractrions
  
  One step equation word problems
  
  Linear inequalities word problems
  
  Ratio and proportion word problems
  
  Time and work word problems
  
  Word problems on sets and venn diagrams
  
  Word problems on ages
  
  Pythagorean theorem word problems
  
  Percent of a number word problems
  
  Word problems on constant speed
  
  Word problems on average speed
  
  Word problems on sum of the angles of a triangle is 180 degree
  
  OTHER TOPICS
  
  Profit and loss shortcuts
  
  Percentage shortcuts
  
  Times table shortcuts
  
  Time, speed and distance shortcuts
  
  Ratio and proportion shortcuts
  
  Domain and range of rational functions
  
  Domain and range of rational functions with holes
  
  Graphing rational functions
  
  Graphing rational functions with holes
  
  Converting repeating decimals in to fractions
  
  Decimal representation of rational numbers
  
  Finding square root using long division
  
  L.C.M method to solve time and work problems
  
  Translating the word problems in to algebraic expressions
  
  Remainder when 2 power 256 is divided by 17
  
  Remainder when 17 power 23 is divided by 16
  
  Sum of all three digit numbers divisible by 6
  
  Sum of all three digit numbers divisible by 7
  
  Sum of all three digit numbers divisible by 8
  
  Sum of all three digit numbers formed using 1, 3, 4
  
  Sum of all three four digit numbers formed with non zero digits
  
  Sum of all three four digit numbers formed using 0, 1, 2, 3
  
  Sum of all three four digit numbers formed using 1, 2, 5, 6
explanation: The document is a worksheet on exponents and square roots, containing a series of problems and their solutions. Each problem is followed by a detailed solution, providing step-by-step explanations.
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distilabel_metadata.content: EXPONENTS AND SQUARE ROOTS WORKSHEET
  
  About "Exponents and Square Roots Worksheet"
  
  Exponents and Square Roots Worksheet :
  
  Worksheet given in this section is much useful to the students who would like to practice problems on exponents and square roots.
  
  Before look at the worksheet, if you would like to learn the stuff exponents and square roots in detail,
  
  Exponents and Square Roots Worksheet - Problems
  
  Problem 1 :
  
  Simplify the following :
  
  (-4)5 ÷ (-4)8
  
  Problem 2 :
  
  If a-1/2  =  5, then find the value of a.
  
  Problem 3 :
  
  If x2  =  y3 and x3z  =  y9, then find the value of z.
  
  Problem 4 :
  
  Solve for k :
  
  3x2 ⋅ 2x3  =  6xk
  
  Problem 5 :
  
  Rationalize the denominator :
  
  / (3 + √2)
  
  Problem 6 :
  
  Rationalize the denominator :
  
  (1 - √5) / (3 + √5)
  
  Problem 7 :
  
  Find the value of :
  
  √2.56
  
  Problem 8 :
  
  Simplify the following square root expression :
  
  (√17)(√51)
  
  Problem 9 :
  
  If x2y3  =  10 and x3y2  =  8, then find the value of x5y5.
  
  Problem 10 :
  
  If (√9)-7 ⋅ (√3)-4  =  3k, then solve for k.
  
  Exponents and Square Roots Worksheet - Solutions
  
  Problem 1 :
  
  Simplify the following :
  
  (-4)5 ÷ (-4)8
  
  Solution :
  
  (-4)5 ÷ (-4) =  (-4)5-8
  
  (-4)5 ÷ (-4) =  (-4)-3
  
  (-4)5 ÷ (-4) =  1 / (-4)3
  
  (-4)5 ÷ (-4) =  1 / (-64)
  
  (-4)5 ÷ (-4) =  - 1 / 64
  
  Problem 2 :
  
  If a-1/2  =  5, then find the value of a.
  
  Solution :
  
  a-1/2  =  5
  
  a  =  5-2/1
  
  a  =  5-2
  
  a  =  1/52
  
  a  =  1/25
  
  Problem 3 :
  
  If x2  =  y3 and x3z  =  y9, then find the value of z.
  
  Solution :
  
  x3z  =  y9
  
  x3z  =  y⋅ 3
  
  x3z  =  (y3)3
  
  Substitute xfor y3.
  
  x3z  =  (x2)3
  
  x3z  =  x6
  
  3z  =  6
  
  Divide each side by 3.
  
  z  =  2
  
  So, the value of z is 2.
  
  Problem 4 :
  
  Solve for k :
  
  3x2 ⋅ 2x3  =  6xk
  
  Solution :
  
  3x2 ⋅ 2x3  =  6xk
  
  (3 ⋅ 2)(x⋅ x3)  =  6xk
  
  (6)(x2+3)  =  6xk
  
  6x5  =  6xk
  
  Divide each side by 6.
  
  x5  =  xk
  
  Using laws of exponents, we have
  
  5  =  k
  
  So, the value of k is 5.
  
  Problem 5 :
  
  Rationalize the denominator :
  
  / (3 + √2)
  
  Solution :
  
  To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √2), that is by (3 - √2).
  
  1 / (3 + √2)  =  [1 ⋅ (3-√2)] / [(3+√2)  (3-√2)]
  
  1 / (3 + √2)  =  (3-√2) / [(3+√2)  (3-√2)]
  
  Using the algebraic identity a2 - b2  =  (a + b)(a - b), simplify the denominator on the right side.
  
  1 / (3 + √2)  =  (3-√2) / [32 - (√2)2]
  
  1 / (3 + √2)  =  (3-√2) / (9 - 2)
  
  1 / (3 + √2)  =  (3 - √2) / 7
  
  Problem 6 :
  
  Rationalize the denominator :
  
  (1 - √5) / (3 + √5)
  
  Solution :
  
  To get rid of the radical in denominator, multiply both numerator and denominator by the conjugate of (3 + √5), that is by (3 - √5).
  
  (1 - √5) / (3 + √5)  =  [(1-√5) ⋅ (3-√5)] / [(3+√5)  (3-√5)]
  
  Simplify.
  
  (1 - √5) / (3 + √5)  =  [3 √5 - 3√5 + 5] / [32 - (√5)2]
  
  (1 - √5) / (3 + √5)  =  (8 - 4√5) / (9 - 5)
  
  (1 - √5) / (3 + √5)  =  4(2 - √5) / 4
  
  (1 - √5) / (3 + √5)  =  2 - √5
  
  Problem 7 :
  
  Find the value of :
  
  √2.56
  
  Solution :
  
  To get rid of the decimal, multiply and divide 2.56 inside the radical by 100.
  
  √2.56  =  √(2.56  100 / 100)
  
  √2.56  =  √(256 / 100)
  
  √2.56   √(16  16) / (10 ⋅ 10)
  
  √2.56  =  16 / 10
  
  √2.56  =  1.6
  
  Problem 8 :
  
  Simplify the following square root expression :
  
  (√17)(√51)
  
  Solution :
  
  Decompose 17 and 51 into prime factors.
  
  Because 17 is a prime number, it can't be decomposed anymore. So, √17 has to be kept as it is.
  
  √51  =  √(3 ⋅ 17)  =  √3 ⋅ √17
  
  So, we have
  
  (√17)(√51)  =  (√17)(√3 ⋅ √17)
  
  (√17)(√51)  =  (√17 ⋅ √17)√3
  
  (√17)(√51)  =  17√3
  
  Problem 9 :
  
  If x2y3  =  10 and x3y2  =  8, then find the value of x5y5.
  
  Solution :
  
  x2y3  =  10 -----(1)
  
  x3y2  =  8 -----(2)
  
  Multiply (1) and (2) :
  
  (1) ⋅ (2) -----> (x2y3) ⋅ (x3y2)  =  10 ⋅ 8
  
  x5y5  =  80
  
  So, the value x5yis 80.
  
  Problem 10 :
  
  If (√9)-7 ⋅ (√3)-4  =  3k, then solve for k.
  
  Solution :
  
  (91/2)-7 ⋅ (31/2)-4  =  3k
  
  (9)-7/2 ⋅ (3)-4/2  =  3k
  
  (32)-7/2 ⋅ 3-2  =  3k
  
  3⋅ (-7/2) ⋅ 3-2  =  3k
  
  3-7 ⋅ 3-2  =  3k
  
  3-7 - 2  =  3k
  
  3-9  =  3k
  
  k  =  -9
  
  So, the value of k is -9.
  
  After having gone through the stuff given above, we hope that the students would have understood "Exponents and Square Roots Worksheet".
  
  Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
  
  You can also visit our following web pages on different stuff in math.
  
  WORD PROBLEMS
  
  Word problems on simple equations
  
  Word problems on linear equations
  
  Algebra word problems
  
  Word problems on trains
  
  Area and perimeter word problems
  
  Word problems on direct variation and inverse variation
  
  Word problems on unit price
  
  Word problems on unit rate
  
  Word problems on comparing rates
  
  Converting customary units word problems
  
  Converting metric units word problems
  
  Word problems on simple interest
  
  Word problems on compound interest
  
  Word problems on types of angles
  
  Complementary and supplementary angles word problems
  
  Double facts word problems
  
  Trigonometry word problems
  
  Percentage word problems
  
  Profit and loss word problems
  
  Markup and markdown word problems
  
  Decimal word problems
  
  Word problems on fractions
  
  Word problems on mixed fractrions
  
  One step equation word problems
  
  Linear inequalities word problems
  
  Ratio and proportion word problems
  
  Time and work word problems
  
  Word problems on sets and venn diagrams
  
  Word problems on ages
  
  Pythagorean theorem word problems
  
  Percent of a number word problems
  
  Word problems on constant speed
  
  Word problems on average speed
  
  Word problems on sum of the angles of a triangle is 180 degree
  
  OTHER TOPICS
  
  Profit and loss shortcuts
  
  Percentage shortcuts
  
  Times table shortcuts
  
  Time, speed and distance shortcuts
  
  Ratio and proportion shortcuts
  
  Domain and range of rational functions
  
  Domain and range of rational functions with holes
  
  Graphing rational functions
  
  Graphing rational functions with holes
  
  Converting repeating decimals in to fractions
  
  Decimal representation of rational numbers
  
  Finding square root using long division
  
  L.C.M method to solve time and work problems
  
  Translating the word problems in to algebraic expressions
  
  Remainder when 2 power 256 is divided by 17
  
  Remainder when 17 power 23 is divided by 16
  
  Sum of all three digit numbers divisible by 6
  
  Sum of all three digit numbers divisible by 7
  
  Sum of all three digit numbers divisible by 8
  
  Sum of all three digit numbers formed using 1, 3, 4
  
  Sum of all three four digit numbers formed with non zero digits
  
  Sum of all three four digit numbers formed using 0, 1, 2, 3
  
  Sum of all three four digit numbers formed using 1, 2, 5, 6
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a worksheet on exponents and square roots, containing a series of problems and their solutions. Each problem is followed by a detailed solution, providing step-by-step explanations.
  
  Label: multiple-question-and-answer