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.
1) To reduce the expression to lowest terms, we need to simplify it as much as possible. Let's break it down step by step:
Step 1: Simplify the numerator
The numerator of the expression is:
(2x + 5) * 4x - 2x^2 * 2
Multiplying out the terms, we get:
8x^2 + 20x - 4x^2
Step 2: Simplify the denominator
The denominator of the expression is:
(2x + 5)^2
Expanding the square, we get:
(2x + 5)(2x + 5) = 4x^2 + 20x + 25
Step 3: Combine the numerator and denominator
Now, we can rewrite the original expression as:
(8x^2 + 20x - 4x^2) / (4x^2 + 20x + 25)
Step 4: Simplify further, if possible
To check if we can simplify any further, let's simplify the numerator and denominator separately and see if they have any common factors:
Numerator: 8x^2 + 20x - 4x^2
Combine like terms: 4x^2 + 20x
Factor out 4x: 4x(x + 5)
Denominator: 4x^2 + 20x + 25
This expression cannot be simplified further.
Step 5: Final Answer
After simplifying, our expression becomes:
(4x(x + 5)) / (4x^2 + 20x + 25)
Therefore, the expression cannot be reduced any further.
2) To simplify the expression, let's break it down step by step:
Step 1: Simplify the numerator
The numerator of the expression is:
(-3 x^4/7) ^4
To raise a power to another power, we multiply the exponents:
(-3) ^ (4 * 4) * (x ^ (4 * 4)) / (7 ^ (4 * 4))
Simplifying further, we have:
(-3) ^ 16 * (x ^ 16) / (7 ^ 16)
Step 2: Simplify the denominator
The denominator of the expression is:
x^-1/2
To simplify negative exponents, we can rewrite them as positive exponents by moving them to the opposite side of the fraction:
1 / (x^(1/2))
Step 3: Combine numerator and denominator
Now, we can combine the numerator and denominator of the expression:
(-3) ^ 16 * (x ^ 16) / (7 ^ 16 * x^(1/2))
Step 4: Simplify further, if possible
To check if we can simplify any further, let's see if there are any common factors between the terms:
For the numerator, (-3) ^ 16 * (x ^ 16), and the denominator, 7 ^ 16 * x^(1/2), there are no common factors to simplify.
Step 5: Final Answer
After simplifying, our expression becomes:
(-3) ^ 16 * (x ^ 16) / (7 ^ 16 * x^(1/2))
Therefore, this is the simplified form of the expression with only positive exponents.
1) To reduce the given expression to its lowest terms, you can follow these steps:
Step 1: Simplify the numerator by expanding and combining like terms.
(2x + 5) * 4x - 2x^2 * 2
8x^2 + 20x - 4x^2
8x^2 - 4x^2 + 20x
4x^2 + 20x
Step 2: Simplify the denominator by squaring the binomial (2x + 5)^2.
(2x + 5)^2 = (2x + 5) * (2x + 5)
(2x + 5)^2 = 4x^2 + 20x + 20x + 25
(2x + 5)^2 = 4x^2 + 40x + 25
Now, we can rewrite the expression in its simplified form:
(4x^2 + 20x) / (4x^2 + 40x + 25)
2) To simplify the given expression and express the answer with only positive exponents, follow these steps:
Step 1: Distribute the exponent outside the bracket to each term inside.
(-3 x^4/7)^4 = (-3^4) * (x^(4/7))^4
(-3 x^4/7)^4 = 81 * (x^(4/7))^4
Step 2: Simplify the constant term.
(-3 x^4/7)^4 = 81 * x^(4/7 * 4)
(-3 x^4/7)^4 = 81 * x^(16/7)
Step 3: Rewrite the exponent with only positive exponents.
(-3 x^4/7)^4 = 81 * (x^(16/7))^1
(-3 x^4/7)^4 = 81 * x^(16/7)
Hence, the simplified expression is 81 * x^(16/7).