I have to simplify this:
(x^(1/2) * z^(-4/5)^4
My work:
x^2/z^(16/5)
Correct?
Simplify (3+5SQRT2)(4+SQRT10)
I have no idea how to foil this one.
To simplify the expression (x^(1/2) * z^(-4/5))^4, you correctly applied the power rule for exponents. However, your final answer is not correct.
Let's go through it step by step:
1. Start with the expression (x^(1/2) * z^(-4/5))^4.
2. Apply the power rule for exponents to each term within the parentheses:
- For x^(1/2), raise both the base (x) and the exponent (1/2) to the power of 4. This gives us x^(1/2 * 4) = x^2.
- For z^(-4/5), raise both the base (z) and the exponent (-4/5) to the power of 4. This gives us z^(-4/5 * 4) = z^(-16/5).
3. Since the expression within the parentheses has been raised to the power of 4, we can now multiply the two terms by applying the product rule for exponents:
- x^2 * z^(-16/5).
4. To simplify further, combine the terms with the same base and apply the quotient rule for exponents:
- x^2 / z^(16/5).
So, the correct simplified form of the expression is x^2 / z^(16/5).
Moving on to the next question:
To simplify the expression (3 + 5√2)(4 + √10), you can use the FOIL method, which stands for First, Outer, Inner, Last. Here's how you can apply it:
1. Start with the expression (3 + 5√2)(4 + √10).
2. Multiply the First terms: 3 * 4 = 12.
3. Multiply the Outer terms: 3 * √10 = 3√10.
4. Multiply the Inner terms: 5√2 * 4 = 20√2.
5. Multiply the Last terms: 5√2 * √10 = 5√(2 * 10) = 5√20 = 10√5.
6. Combine the results from steps 2-5: 12 + 3√10 + 20√2 + 10√5.
7. Finally, simplify if possible.
The result cannot be further simplified unless there are like terms that can be combined. So, the final answer is 12 + 3√10 + 20√2 + 10√5.