1) An expression that occurs in calculus in given. Reduce the expression to lowest terms.
(2x + 5) 4x - 2x^2(2) / (2x+5)^2
I have no idea what to do for this.
2) Simplify the expression. Express the answer so that only positive exponents occur. Assume that all variables are positive.
(-3 x^4/7) ^4 / x^-1/2
I know that (-3)^4 would be 81 but I do not know what else to do.
on the first, I have no idea what the numerator is.
On the second
(-3)^4 (x^4/7)^4 (x^1/2)=
81* x^(16/7 + 1/2)
YOu can combine that x exponent.
1) To reduce the given expression to its lowest terms, you can proceed with the following steps:
Step 1: Expand the numerator:
(2x + 5) * 4x - 2x^2 * (2)
This gives us: 8x^2 + 20x - 4x^2
Step 2: Expand the denominator:
(2x + 5)^2
This gives us: (2x + 5) * (2x + 5) = (2x)^2 + 2 * (2x) * 5 + 5^2 = 4x^2 + 20x + 25
Now, the original expression can be written as:
(8x^2 + 20x - 4x^2) / (4x^2 + 20x + 25)
Step 3: Combine like terms in the numerator and denominator:
(8x^2 - 4x^2 + 20x) / (4x^2 + 20x + 25)
Simplifying further:
4x^2 + 20x / (4x^2 + 20x + 25)
Step 4: Factor out any common terms in the numerator and denominator:
4x (x + 5) / (4x^2 + 20x + 25)
And that is the expression in its simplest form.
2) To simplify the given expression and express the answer with only positive exponents, we can follow these steps:
Step 1: Apply the power of a power rule to the numerator:
(-3 x^4/7)^4 = (-3)^4 * (x^4/7)^4
This gives us: 81 * (x^4/7)^4
Step 2: Apply the power of a power rule to the denominator:
x^(-1/2) = 1 / x^(1/2)
Step 3: Combine the numerator and denominator to obtain the simplified expression:
81 * (x^4/7)^4 / (1 / x^(1/2))
To simplify even further, we can apply the reciprocal property by multiplying the numerator by the reciprocal of the denominator:
81 * (x^4/7)^4 * x^(1/2)
To simplify the expressions within parentheses, apply the power of a power rule again:
81 * x^(16/7) * x^(1/2)
Now, apply the product of powers rule by adding the exponents of x:
81 * x^(16/7 + 1/2) = 81 * x^(16/7 + 7/14) = 81 * x^(16/7 + 1/2)
To combine the fractions, find a common denominator (in this case, 14):
81 * x^((32 + 7)/14) = 81 * x^(39/14)
This is the final simplified expression with only positive exponents.