Evaluate square root of 1/((1/(14910^9))^2-(1/(29810^9))^2)

$\sqrt{\frac{1}{{(\frac{1}{149 \cdot 10^{9}})}^{2} - {(\frac{1}{298 \cdot 10^{9}})}^{2}}}$

Step 1

Divide using scientific notation.

Group coefficients together and exponents together to divide numbers in scientific notation.

$\sqrt{\frac{1}{{((\frac{1}{149}) (\frac{1}{10^{9}}))}^{2} - {(\frac{1}{298 \cdot 10^{9}})}^{2}}}$

Divide $1$ by $149$ .

$\sqrt{\frac{1}{{(0.0067114 \frac{1}{10^{9}})}^{2} - {(\frac{1}{298 \cdot 10^{9}})}^{2}}}$

Move $10^{9}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\sqrt{\frac{1}{{(0.0067114 \cdot 10^{- 9})}^{2} - {(\frac{1}{298 \cdot 10^{9}})}^{2}}}$

Step 2

Divide using scientific notation.

Group coefficients together and exponents together to divide numbers in scientific notation.

$\sqrt{\frac{1}{{(0.0067114 \cdot 10^{- 9})}^{2} - {((\frac{1}{298}) (\frac{1}{10^{9}}))}^{2}}}$

Divide $1$ by $298$ .

$\sqrt{\frac{1}{{(0.0067114 \cdot 10^{- 9})}^{2} - {(0.0033557 \frac{1}{10^{9}})}^{2}}}$

Move $10^{9}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\sqrt{\frac{1}{{(0.0067114 \cdot 10^{- 9})}^{2} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 3

Move the decimal point in $0.0067114$ to the right by $3$ places and decrease the power of $10^{- 9}$ by $3$ .

$\sqrt{\frac{1}{{(6.71140939 \cdot 10^{- 12})}^{2} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 4

Apply the product rule to $6.71140939 \cdot 10^{- 12}$ .

$\sqrt{\frac{1}{{6.71140939}^{2} \cdot {(10^{- 12})}^{2} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 5

Raise $6.71140939$ to the power of $2$ .

$\sqrt{\frac{1}{45.04301608 \cdot {(10^{- 12})}^{2} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 6

Multiply the exponents in ${(10^{- 12})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{1}{45.04301608 \cdot 10^{- 12 \cdot 2} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Multiply $- 12$ by $2$ .

$\sqrt{\frac{1}{45.04301608 \cdot 10^{- 24} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 7

Move the decimal point in $45.04301608$ to the left by $1$ place and increase the power of $10^{- 24}$ by $1$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - {(0.0033557 \cdot 10^{- 9})}^{2}}}$

Step 8

Move the decimal point in $0.0033557$ to the right by $3$ places and decrease the power of $10^{- 9}$ by $3$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - {(3.35570469 \cdot 10^{- 12})}^{2}}}$

Step 9

Apply the product rule to $3.35570469 \cdot 10^{- 12}$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - ({3.35570469}^{2} \cdot {(10^{- 12})}^{2})}}$

Step 10

Raise $3.35570469$ to the power of $2$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - (11.26075402 \cdot {(10^{- 12})}^{2})}}$

Step 11

Multiply the exponents in ${(10^{- 12})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - (11.26075402 \cdot 10^{- 12 \cdot 2})}}$

Multiply $- 12$ by $2$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - 1 \cdot 11.26075402 \cdot 10^{- 24}}}$

Step 12

Multiply $- 1$ by $11.26075402$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - 11.26075402 \cdot 10^{- 24}}}$

Step 13

Move the decimal point in $- 11.26075402$ to the left by $1$ place and increase the power of $10^{- 24}$ by $1$ .

$\sqrt{\frac{1}{4.5043016 \cdot 10^{- 23} - 1.1260754 \cdot 10^{- 23}}}$

Step 14

Subtract $1.1260754 \cdot 10^{- 23}$ from $4.5043016 \cdot 10^{- 23}$ .

$\sqrt{\frac{1}{3.3782262 \cdot 10^{- 23}}}$

Step 15

Divide using scientific notation.

Group coefficients together and exponents together to divide numbers in scientific notation.

$\sqrt{(\frac{1}{3.3782262}) (\frac{1}{10^{- 23}})}$

Divide $1$ by $3.3782262$ .

$\sqrt{0.29601333 \frac{1}{10^{- 23}}}$

Move $10^{- 23}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt{0.29601333 \cdot 10^{23}}$

Step 16

Move the decimal point in $0.29601333$ to the right by $1$ place and decrease the power of $10^{23}$ by $1$ .

$\sqrt{2.96013333 \cdot 10^{22}}$

Step 17

Raise $10$ to the power of $22$ .

$\sqrt{2.96013333 \cdot 10000000000000000000000}$

Step 18

Multiply $2.96013333$ by $10000000000000000000000$ .

$\sqrt{29601333333333294354297.6}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\sqrt{29601333333333294354297.6}$

Decimal Form:

$172050380218.508$

Evaluate square root of 1/((1/(149*10^9))^2-(1/(298*10^9))^2)

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Evaluate square root of 1/((1/(14910^9))^2-(1/(29810^9))^2)