**Exponent properties and scientific notation worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on exponent properties and scientific notation.

**Problem 1 :**

Simplify the following expression :

(2x^{5})^{4}(x^{3})^{2}

**Problem 2 :**

Simplify the following expression :

(2x^{-3})^{3} (3x^{3})^{2}

**Problem 3 :**

Simplify the following expression :

(2x^{3})^{2} (a^{0})^{5}

**Problem 4 :**

Simplify the following expression :

(3x^{6}/2y^{9}) ⋅ (y^{5}/x^{3})

**Problem 5 :**

Simplify the following expression :

(2x^{2}y^{-3})(3x^{-4}y^{-2})

**Problem 6 :**

Archimedes estimated the universe to be 2.3x10^{19 }millimeters (mm) in diameter. Write this number in standard form.

**Problem 7 :**

Write the following number in scientific notation.

(2 x 10^{5})(4 x 10^{4})

**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}, -n^{3 }

**Problem 1 :**

Simplify the following expression :

(2x^{5})^{4}(x^{3})^{2}

**Solution :**

= (2x^{5})^{4} (x^{3})^{2}

By distributing the power, we get

= 2^{4}(x^{5})^{4} (x^{3})^{2}

= 16x^{20} x^{6}

= 16x^{20 + 6}

= 16x^{26}

**Problem 2 :**

Simplify the following expression :

(2x^{-3})^{3} (3x^{3})^{2}

**Solution :**

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

By distributing the power, we get

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

= 8x^{-9} ⋅ 9x^{6}

= (8 ⋅ 9)x^{-9 + 6}

= 72x^{-3}

= 72/x^{3}

**Problem 3 :**

Simplify the following expression :

(2x^{3})^{2} (a^{0})^{5}

**Solution :**

= (2x^{3})^{2} (a^{0})^{5}

By distributing the power, we get

= 2^{2}(x^{3})^{2} (1)^{5}

= 4x^{6} (1)

= 4x^{6}

**Problem 4 :**

Simplify the following expression :

(3x^{6}/2y^{9}) ⋅ (y^{5}/x^{3})

**Solution :**

= (3x^{6}/2y^{9}) ⋅ (y^{5}/x^{3})

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

= (3x^{6}⋅x^{-3) }/ (2y^{9}⋅ y^{-5})

= 3x^{(6 - 3) }/ 2y^{(9 - 5)}

= 3x^{3 }/ 2y^{4}

**Problem 5 :**

Simplify the following expression :

(2x^{2}y^{-3})(3x^{-4}y^{-2})

**Solution :**

= (2x^{2}y^{-3}) (3x^{-4}y^{-2})

= (2 ⋅ 3)x^{2-4 }y^{-3 - 2}

= 6x^{-2 }y^{-5}

= 6/x^{2 }y^{5}

**Problem 6 :**

Archimedes estimated the universe to be 2.3x10^{19 }millimeters (mm) in diameter. Write this number in standard form.

**Solution :**

In the number 2.3x10^{19 }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 10^{19 } = 23000000000000000000

**Problem 7 :**

Write the following number in scientific notation.

(2 x 10^{5})(4 x 10^{4})

**Solution :**

= (2 x 10^{5})(4 x 10^{4})

= 2 x 4 x 10^{5} x 10^{4}

= 8 x 10^{5 + 4}

= 8 x 10^{9}

**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}, -n^{3 }

**Solution :**

If n > 0, then n is positive.

(i) -n^{-3 } = -1/n^{3 }(Negative)

(ii) n^{-3 } = 1/n^{3 }(Positive)

(iii) (-n)^{-3 } = 1/(-n)^{3} = 1/(-n^{3}) = -1/n^{3 }(Negative)

(iv) (-n)^{3 }= -n^{3} (Negative)

(v) -n^{3 } (Negative)

After having gone through the stuff given above, we hope that the students would have understood how to do problems on exponent properties and scientific notation.

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