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
The text passage is a worksheet containing problems and solutions related to exponents and square roots. Each problem is followed by a detailed solution.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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
user
The text passage is a worksheet containing problems and solutions related to exponents and square roots. Each problem is followed by a detailed solution.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS