Question

Solve $\dfrac{{(6.7 \times {{10}^{ - 11}})(6 \times {{10}^{24}})(7.4 \times {{10}^{22}})}}{{{{(3.84 \times {{10}^8})}^2}}}$

Answer 1

Hint: In mathematical expression, the power is used to express the long expression in the short form which is used generally that represents the repeated multiplication of the same factor. Here we will use the negative exponent rule to simplify the given expression.

Complete step-by-step answer:
Take the given expression: $\dfrac{{(6.7 \times {{10}^{ - 11}})(6 \times {{10}^{24}})(7.4 \times {{10}^{22}})}}{{{{(3.84 \times {{10}^8})}^2}}}$
When there is a whole square to the terms, then the square is applied to both the terms inside the bracket. Square can be defined as the term multiplied with itself.
$ = \dfrac{{(6.7 \times {{10}^{ - 11}} \times 6 \times {{10}^{24}} \times 7.4 \times {{10}^{22}}}}{{(3.84 \times 3.84 \times {{10}^8} \times {{10}^8})}}$
When the terms are in multiplication with the same base then the powers are added.
$ = \dfrac{{(6.7 \times 6 \times 7.4 \times {{10}^{24 + 22 - 11}}}}{{(3.84 \times 3.84 \times {{10}^{8 + 8}})}}$
Simplify the above expression, adding the powers and exponents.
$ = \dfrac{{6.7 \times 6 \times 7.4 \times {{10}^{35}}}}{{3.84 \times 3.84 \times {{10}^{16}}}}$
Find the multiplication of the terms and use the negative exponent rule, By using the law of the negative exponent rule which states that when the power and exponent moved to the denominator negative power becomes positive that is ${a^{ - n}} = \dfrac{1}{{{a^n}}}$
$ = \dfrac{{6.7 \times 6 \times 7.4 \times {{10}^{35 - 16}}}}{{3.84 \times 3.84}}$
Simplify by finding the difference of powers in the numerator.
$ = \dfrac{{6.7 \times 6 \times 7.4 \times {{10}^{19}}}}{{3.84 \times 3.84}}$
Find the product of the terms in the above expression –
$ = \dfrac{{297.84 \times {{10}^{19}}}}{{14.7456}}$
Find the division –
$ = 20.19856 \times {10^{19}}$
This is the required expression.
So, the correct answer is “Option B”.

Note: Remember the seven basic rules of the exponent or the laws of exponents to solve these types of examples. Make sure to go through the below mentioned rules which defines how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
I.Product of powers rule
II.Quotient of powers rule
III.Power of a power rule
IV.Power of a product rule
V.Power of a quotient rule
VI.Zero power rule
VII.Negative exponent rule