**Exponent properties and scientific notation worksheet :**

Here we are going to see some practice questions on exponent properties and scientific notation.

**Question 1 :**

Simplify the following expression

**Solution :**

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

By distributing the power, we get

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

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

Since we have same base, we need to write only one base and combine the powers according to their signs.

= 16 x^{(20 + 6)}

= 16 x^{26}

**Question 2 :**

Simplify the following expression

**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⋅ 9x^{(-9 + 6)}

= 72x^{-3}

= 72/x^{3}

**Question 3 :**

Simplify the following expression

**Solution :**

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

By distributing the power, we get

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

= 4 x^{6} (1)

= 4 x^{6}

**Question 4 :**

Simplify the following expression

**Solution :**

= (3x^{6}/2y^{9}) ⋅ (y^{5}/x^{3})

Here we need to combine the x and y terms.Let us combine the x terms in the numerator and y terms in the denominator.

= (3x^{6}⋅x^{3)}/(2y^{9}⋅ y^{5})

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

= 3x^{9}/2y^{14}

**Question 5 :**

Simplify the following expression

**Solution :**

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

= 2⋅3 x^{(2-4) }y^{(-3 - 2)}

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

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

**Question 6 :**

Archimedes estimated the universe to be 2.3 x 10^{19} millimeters (mm) in diameter.
If this number were written in standard or decimal form, how many 0s would
follow the digit 3?

**Solution :**

To write 2.3 x 10^{19} in standard form, we have to move the decimal point 19 digits. Since we have only one number after the decimal point we have to write 18 zeroes.

2.3 x 10^{19 } = 23000000000000000000

**Question 7 :**

perform the indicated calculations. Write your result 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}

**Question 7 :**

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)

**Question 7 :**

If n > 0, which of the following expressions are negative? -n^{-3} , n^{-3} , (-n)^{-3} , (-n)^{3} , -n^{3 }

**Solution :**

n > 0 means, n is a positive

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

Though we have changed it as positive exponent, it is negative

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

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

(iv) (-n)^{3 }

By distributing the power, we get

= -n^{3} (Negative)

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

After having gone through the stuff given above, we hope that the students would have understood, "Exponent properties and scientific notation worksheet".

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