What is the value of the expression below? \frac{\sqrt[3]{2.7 \times 10^7} \cdot \sqrt{1.6… — ACT

What is the value of the expression below?

\frac{\sqrt[3]{2.7 \times 10^7} \cdot \sqrt{1.6 \times 10^9}}{\sqrt{2.5 \times 10^{-1}}}

The correct value of the expression is

2.4 \times 10^7

Evaluating the terms step-by-step, the first term in the numerator simplifies to

3 \times 10^2

and the second term simplifies to

4 \times 10^4

. Multiplying these values results in

1.2 \times 10^7

. The denominator evaluates to

0.5

. Dividing

1.2 \times 10^7

0.5

yields

2.4 \times 10^7

Simplify the first term in the numerator:

\sqrt[3]{2.7 \times 10^7}

3 \times 10^2

Rewrite

2.7 \times 10^7

27 \times 10^6

. Since 27 is a perfect cube (

3^3

) and

10^6

is a perfect cube (

(10^2)^3

), the cube root is

3 \times 10^2

Simplify the second term in the numerator:

\sqrt{1.6 \times 10^9}

4 \times 10^4

Rewrite

1.6 \times 10^9

16 \times 10^8

. Since 16 is a perfect square (

4^2

) and

10^8

is a perfect square (

(10^4)^2

), the square root is

4 \times 10^4

Multiply the simplified numerator terms.

1.2 \times 10^7

Multiply the coefficients and add the exponents:

(3 \times 10^2) \times (4 \times 10^4) = (3 \times 4) \times 10^{2+4} = 12 \times 10^6 = 1.2 \times 10^7

Simplify the denominator:

\sqrt{2.5 \times 10^{-1}}

5 \times 10^{-1}

(or

0.5

)

Rewrite

2.5 \times 10^{-1}

0.25

, or as

25 \times 10^{-2}

. The square root is

\sqrt{25} \times \sqrt{10^{-2}} = 5 \times 10^{-1} = 0.5

Divide the numerator by the denominator.

2.4 \times 10^7

Divide

1.2 \times 10^7

0.5

to get

2.4 \times 10^7

. In scientific notation terms:

\frac{1.2 \times 10^7}{5 \times 10^{-1}} = 0.24 \times 10^8 = 2.4 \times 10^7

Evaluating calculations involving square and cube roots of scientific notation expressions by adjusting decimal places to construct perfect powers.

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