Simplify each expression (where possible) and write the answer without using negative or fractional exponents. Assume that X>0.
a. 8x^(1/3)
b. (-8x)^(1/3)
c. (8x)^(-1/3)
d. (3x^(3/4))(16x)^(1/4)
e. (x^(1/2)) / (x^(5/2)
so far my answers are : a.2x b. -2x c.1/2x but i feel that i'm doing it completely wrong. d&e i don't know how to solve them...
Could someone explain how to simplify them? Thank You!
a.
If it is 8 * x^(1/3)
that can not be simplifies.
If it were (8 x)^(1/3)
that would be 2 x^(1/3)
b. -8(1/3) = -2 yes
but x^(1/3) = x^(1/3), no simplification
so I get -2 x^(1/3)
c. 1/8^(1/3) = 1/2
so I get
(1/2)(x^-(1/3) )
or
1/[2 x^(1/3)]
d. 3 * 16^(1/4) * x^(3/4)*x^(1/4)
3*2*x^1
6 x
e. x^(1/2 - 5/2) = x^-4/2 = x^-2 = 1/x^2
To simplify expressions with fractional exponents, we can use the rules of exponents. Let's go through each expression:
a. 8x^(1/3):
To simplify, we use the rule that says for any positive number "a" and any integer "m," a^(1/m) equals the mth root of "a." So, 8x^(1/3) is the same as the cube root of (8x).
b. (-8x)^(1/3):
In this case, we have a negative number raised to the 1/3 power. To deal with this, we first need to evaluate the cube root of (-8x) and then apply the negative sign afterward.
c. (8x)^(-1/3):
The negative exponent can be handled by using the rule that says for any number "a," a^(-n) is equal to 1/a^n. So, (8x)^(-1/3) is the same as 1 / (8x)^(1/3).
d. (3x^(3/4))(16x)^(1/4):
To simplify this expression, we can multiply the coefficients (3 and 16) and then multiply the variables with the same base (x). For the variable x, we apply the rule that (x^a)(x^b) equals x^(a+b), which means we add the exponents.
e. (x^(1/2)) / (x^(5/2)):
In this case, we have the division of two terms with the same base, x. The rule to simplify powers with the same base when dividing is to subtract the exponent in the denominator from the exponent in the numerator.
The simplified answers for each expression are as follows:
a. 8x^(1/3) simplifies to 2x.
b. (-8x)^(1/3) simplifies to -2x.
c. (8x)^(-1/3) simplifies to 1 / 2x^(1/3).
d. (3x^(3/4))(16x)^(1/4) simplifies to 12x^(1).
e. (x^(1/2)) / (x^(5/2)) simplifies to 1 / x.