Question

1. write the following in decimal notation: 7.388×10^{10}

738800000000
0.0000000007388
484472207.006453
0.000000000007388
73880000000000

1. if the geometric mean of 8 and x is 12, calculate the value of x.

x =

1. all quadrilaterals with congruent diagonals are rectangles.

true
false

1. find the value of the expression.

-\sqrt{-64}
\sqrt{x}

Explanation:

Step1: Convert scientific - notation to decimal

For a number in scientific notation \(a\times10^{n}\) where \(a = 7.388\) and \(n = 10\), we move the decimal point \(n\) places to the right. So \(7.388\times10^{10}=73880000000\).

Step2: Use geometric - mean formula

The geometric mean of two numbers \(a\) and \(b\) is \(\sqrt{ab}\). Given \(a = 8\), \(b=x\) and the geometric mean is \(12\), we have \(\sqrt{8x}=12\). Squaring both sides gives \(8x = 144\), then \(x=\frac{144}{8}=18\).

Step3: Analyze the property of quadrilaterals

Not all quadrilaterals with congruent diagonals are rectangles. For example, isosceles trapezoids have congruent diagonals but are not rectangles. So the statement is false.

Step4: Evaluate the square - root expression

The square root of a negative number in the real - number system is not defined. But if we consider the complex - number system, \(\sqrt{-64}=8i\), so \(-\sqrt{-64}=- 8i\).

Answer:

1. \(73880000000\)
2. \(18\)
3. False
4. \(-8i\)