**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,

**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 x^{2} = y^{3} and x^{3z} = y^{9}, then find the value of z.

**Problem 4 :**

Solve for k :

3x^{2} ⋅ 2x^{3} = 6x^{k}

**Problem 5 : **

Rationalize the denominator :

1 / (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 x^{2}y^{3} = 10 and x^{3}y^{2} = 8, then find the value of x^{5}y^{5}.

**Problem 10 :**

If (√9)^{-7} ⋅ (√3)^{-4} = 3^{k}, then solve for k.

**Problem 1 :**

Simplify the following :

(-4)^{5} ÷ (-4)^{8}

**Solution :**

(-4)^{5} ÷ (-4)^{8 } = (-4)^{5-8}

(-4)^{5} ÷ (-4)^{8 } = (-4)^{-3}

(-4)^{5} ÷ (-4)^{8 } = 1 / (-4)^{3}

(-4)^{5} ÷ (-4)^{8 } = 1 / (-64)

(-4)^{5} ÷ (-4)^{8 } = - 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 = 1/5^{2}

a = 1/25

**Problem 3 :**

If x^{2} = y^{3} and x^{3z} = y^{9}, then find the value of z.

**Solution :**

x^{3z} = y^{9}

x^{3z} = y^{3 }^{⋅ 3}

x^{3z} = (y^{3})^{3}

Substitute x^{2 }for y^{3}.

x^{3z} = (x^{2})^{3}

x^{3z} = x^{6}

3z = 6

Divide each side by 3.

z = 2

So, the value of z is 2.

**Problem 4 :**

Solve for k :

3x^{2} ⋅ 2x^{3} = 6x^{k}

**Solution : **

3x^{2} ⋅ 2x^{3} = 6x^{k}

(3 ⋅ 2)(x^{2 }⋅ x^{3}) = 6x^{k}

(6)(x^{2+3}) = 6x^{k}

6x^{5} = 6x^{k}

Divide each side by 6.

x^{5} = x^{k}

Using laws of exponents, we have

5 = k

So, the value of k is 5.

**Problem 5 : **

Rationalize the denominator :

1 / (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 a^{2} - b^{2} = (a + b)(a - b), simplify the denominator on the right side.

1 / (3 + √2) = (3-√2) / [3^{2} - (√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] / [3^{2} - (√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 x^{2}y^{3} = 10 and x^{3}y^{2} = 8, then find the value of x^{5}y^{5}.

**Solution : **

x^{2}y^{3} = 10 -----(1)

x^{3}y^{2} = 8 -----(2)

Multiply (1) and (2) :

(1) ⋅ (2) -----> (x^{2}y^{3}) ⋅ (x^{3}y^{2}) = 10 ⋅ 8

x^{5}y^{5} = 80

So, the value x^{5}y^{5 }is 80.

**Problem 10 :**

If (√9)^{-7} ⋅ (√3)^{-4} = 3^{k}, then solve for k.

**Solution :**

(9^{1/2})^{-7} ⋅ (3^{1/2})^{-4} = 3^{k}

(9)^{-7/2} ⋅ (3)^{-4/2} = 3^{k}

(3^{2})^{-7/2} ⋅ 3^{-2} = 3^{k}

3^{2 }^{⋅ (-7/2)} ⋅ 3^{-2} = 3^{k}

3^{-7} ⋅ 3^{-2} = 3^{k}

3^{-7 - 2} = 3^{k}

3^{-9} = 3^{k}

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.

Widget is loading comments...

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 **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**