EXPONENT PROPERTIES AND SCIENTIFIC NOTATION WORKSHEET

Problem 1 :

Simplify the following expression : 

(2x⁵)⁴(x³)²

Problem 2 :

Simplify the following expression :

(2x^-3)³ (3x³)²

Problem 3 :

Simplify the following expression : 

(2x³)² (a⁰)⁵

Problem 4 :

Simplify the following expression :

(3x⁶/2y⁹) ⋅ (y⁵/x³)

Problem 5 :

Simplify the following expression : 

(2x²y^-3)(3x^-4y^-2)

Problem 6 :

Archimedes estimated the universe to be 2.3x10¹⁹ millimeters (mm) in diameter. Write this number in standard form. 

Problem 7 :

Write the following number in scientific notation.

(2 x 10⁵)(4 x 10⁴)

Problem 8 :

Can (a + b)^-1 be written as (1/a) + (1/b)  by using the properties of exponents? If not, why not? Explain.

Problem 9 :

If n > 0, which of the following expressions are positive and negative ?

-n^-3, n^-3, (-n)^-3, (-n)³, -n³  

Detailed Answer Key

Problem 1 :

Simplify the following expression : 

(2x⁵)⁴(x³)²

Solution :

 =  (2x⁵)⁴ (x³)²

By distributing the power, we get

=  2⁴(x⁵)⁴ (x³)²

=  16x²⁰ x⁶

=  16x^{20 + 6}

=  16x²⁶

Problem 2 :

Simplify the following expression :

(2x^-3)³ (3x³)²

Solution :

 =  (2x^-3)³ (3x³)²

By distributing the power, we get

 =  2³(x^-3)³ 3²(x³)²

 =  8x^-9 ⋅ 9x⁶

 =  (8 ⋅ 9)x^{-9 + 6}

 =  72x^-3 

=  72/x³

Problem 3 :

Simplify the following expression : 

(2x³)² (a⁰)⁵

Solution :

 =  (2x³)² (a⁰)⁵

By distributing the power, we get

   =  2²(x³)² (1)⁵

   =  4x⁶ (1)

   =  4x⁶

Problem 4 :

Simplify the following expression :

(3x⁶/2y⁹) ⋅ (y⁵/x³)

Solution :

 =  (3x⁶/2y⁹) ⋅ (y⁵/x³)

Here, we have to combine the x and y terms. 

   =  (3x⁶⋅x^-3) / (2y⁹⋅ y^-5)

   =  3x^{(6 - 3)} / 2y^{(9 - 5)}

   =  3x³ / 2y⁴

Problem 5 :

Simplify the following expression : 

(2x²y^-3)(3x^-4y^-2)

Solution :

 =  (2x²y^-3) (3x^-4y^-2)

 =  (2 ⋅ 3)x^2-4 y^{-3 - 2}

 =  6x^-2 y^-5

 =  6/x² y⁵

Problem 6 :

Archimedes estimated the universe to be 2.3x10¹⁹ millimeters (mm) in diameter. Write this number in standard form. 

Solution :

In the number 2.3x10¹⁹ the exponent of 10 is positive. So, we have to move the decimal point 19 digits to the right. 

In 2.3, we have only one digit after the decimal point. So we have to add eighteen zeros to move the point 19 digits to the right. 

Then, 

2.3 x 10¹⁹  =  23000000000000000000

Problem 7 :

Write the following number in scientific notation.

(2 x 10⁵)(4 x 10⁴)

Solution :

  =  (2 x 10⁵)(4 x 10⁴)

  =  2 x 4 x 10⁵ x 10⁴

  =  8 x 10^{5 + 4}

  =  8 x 10⁹

Problem 8 :

Can (a + b)^-1 be written as (1/a) + (1/b)  by using the properties of exponents? If not, why not? Explain.

Solution :

In order to convert the power as positive, we need to write its reciprocal.

That is,

(a + b)^-1  =  1/(a + b)

But, 1/(a + b) is not equal to (1/a) + (1/b). 

So, (a + b)^-1 can not be written as (1/a) + (1/b). 

Problem 9 :

If n > 0, which of the following expressions are positive and negative ?

-n^-3, n^-3, (-n)^-3, (-n)³, -n³  

Solution :

If n > 0, then n is positive.

(i)   -n^-3  =  -1/n³ (Negative)

(ii)  n^-3  =  1/n³ (Positive)

(iii)  (-n)^-3  =  1/(-n)³  =  1/(-n³)  =  -1/n³ (Negative)

(iv)   (-n)³  =  -n³ (Negative)

(v) -n³  (Negative)

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