Question

HELP!!! I will give a medal!!
Part A: Use the properties of exponents to explain why 81/3 is called the cube root of 8. (5 points)

Part B: The length of a rectangle is 3 units and its width is √3 unit. Is the area of the rectangle rational or irrational? Justify your answer. (5 points)

Answer 1

@whpalmer4

Answer 2

@Nnesha

Answer 3

Anyone?

Answer 4

are u there ???

Answer 5

@katienicole sorry, have been away from OS for a while

Use properties of exponents to explain why \(8^{1/3} = \sqrt[3]{8}\)

\[x^a * x^b = x^{a+b}\]

This means that if \(a=b\) then \[x^a*x^a = x^{a+a} = x^{2a}\]

but remember that the square root of a positive number \(x\) is the number \(u\) such that when multiplied by itself gives you the first number:

\[u=\sqrt{x}\]\[u*u= u^2 = \sqrt{x}*\sqrt{x} = \sqrt{x*x} =x\]

That implies that for square roots, the fractional exponent form would be \(x^{1/2}\) because the sum of the exponents must be \(1\) when we multiply them together, and the only fraction \(a\) which works is

\[a+a = 1\]\[2a=1\]\[a=\frac{1}{2}\]

So \[\sqrt{x} = x^{1/2}\] because \[x^{1/2}*x^{1/2} = x^{\frac{1}{2} + \frac{1}{2}} = x^1 = x\]

You should be able to figure out what this means for cube roots, I hope...

Answer 6

:O :O