Part A: Use the properties of exponents to explain why is called the fifth root of 18.
1. What property refers to roots and fractional powers?
My Answer: Rule of exponents
2. How does it apply to 18^1/5?
If you multiply 18^1/5 by itself 5 times, you get 18.
Part B: The length of a rectangle is 2 units and its width is √2 unit. Is the area of the rectangle rational or irrational? Justify your answer.
1. What is the formula of the area of a rectangle? L x W
2. Plug and Chug your answer: 2x√2 2√2.
3) Is the answer rational or irrational? Irrational
1 and 2 look fine
There is nothing you can multiply sqrt 2 by to get a rational number but sqrt 2 or a multiple thereof
L * W yes
4*2 x = 8 x
as we said above if you multiply sqrt 2 by sqrt 2 it comes out a perfectly rational 2
HOWEVER, you better do it symbolically with sqrt 2, not a calculation rounded off.
I'm confused what you mean by 3? I was thinking Irrational because you can not write as a ratio.??
sqrt 2 * sqrt 2 = 2 = 2/1
you can not write sqrt 2 as a ratio of integers
but you certainly can write 2 as a ratio
and 2 is sqrt 2 * sqrt 2 by definition
However do not try to write
1.41 etc * 1.41 etc as a ratio
Part A:
1. The property that refers to roots and fractional powers is the Rule of Exponents. This property helps us simplify expressions involving radical notation and fractional exponents.
2. In the case of 18^1/5, we can use the Rule of Exponents to understand why it is called the fifth root of 18. The property states that for any real number a and any positive integer n, (a^n)^1/n = a. Applying this property, we find that (18^1/5)^5 = 18. In other words, if we raise 18 to the exponent 1/5 and then raise the result to the exponent 5, it gives us back the original number 18. This demonstrates that 18^1/5 is indeed the fifth root of 18.
Part B:
1. The formula for the area of a rectangle is given by length multiplied by width, or A = L x W.
2. Given that the length is 2 units and the width is √2 units, we can calculate the area by multiplying these values: A = 2 x √2.
3. To determine whether the area of the rectangle is rational or irrational, we need to evaluate the expression 2 x √2. The square root of 2 (√2) is an irrational number because it cannot be expressed as a fraction or terminated decimal. Since we are multiplying 2, a rational number, by √2, an irrational number, the resulting product is irrational. Therefore, the area of the rectangle is irrational.