Maths help with exponentials [closed]

Closed. This question is off-topic. It is not currently accepting answers.

---

Want to improve this question? Update the question so it's on-topic for Stack Overflow.

Closed 11 years ago.

Improve this question

Evaluate

(z x^-1 y)^5 y^5

~~~~~~~~~~~~~~~~~~~~~~~ OVER

x^-4 z^-4

How w开发者_JAVA技巧ould I evaluate this if X = 10, y = -3 and z = 3? I would like a step-by-step solution to help me fully understand it.

---

Numerator evaluates as (z*y*x^-1)^5 * y^5

further rewriting ((z^5*y^5)*y^5)/x^5

Denominator ((1/x^4)*(1/z^4))

Final Answer would be ((y^10)*(z^9))/x

as per your values it (3^19)/10

---

Exponentials have higher priority in most computer languages, so adding parentheses like this should make it clearer. I'm assuming that you're dividing the first polynomial by the second. It's simple algebra.

(z x^-1 y)^5 y^5
---------------- = 
x^-4 z^-4

(y^10)(z^9)/x

You substitute the numbers.

---

Start with:

((z * x^-1 * y)^5 * y^5)/(x^-4 * z^-4)

Commute the exponent to the z factor: (A * B)^N => A^N * B^N

(z^5 * (x^-1 * y)^5 * y^5)/(x^-4 * z^-4)

Commute the exponent to the x and y factors: (A * B)^N => A^N * B^N

(z^5 * (x^-1)^5 * y^5 * y^5)/(x^-4 * z^-4)

Simplify the exponenet on the x factor: (A^N)^M => A^(N*M)

(z^5 * x^-5 * y^5 * y^5)/(x^-4 * z^-4)

Combine the y factors: A^N * A^M => A^(N+M)

(z^5 * x^-5 * y^10)/(x^-4 * z^-4)

Remove the negative exponent on x: 1/A^-N => A^N

(z^5 * x^-5 * y^10 * x^4) / (z^-4)

Remove the negative exponenet on z: 1/A^-N => A^N

z^5 * x^-5 * y^10 * x^4 * z^4

Combine the z factors: A^N * A^M => A^(N+M)

z^9 * x^-5 * y^10 * x^4

Combine the x factors: A^N * A^M => A^(N+M)

z^9 * x^-1 * y^10

Remove the negative exponent on x: A^(-N) => 1/A^N

(z^9 * y^10)/(x^1)

Simplify the x factor: A^1 => A

(z^9 * y^10)/(x)

And that's the algebraic form of your answer.

Next, subsitute the values:

3^9 * (-3)^10 / 10

Factor the exponents:

(3^3)^3 * (-3)^10 / 10

(3^3)^3 * ((-3)^2)^5 / 10

Evalutate the innermost exponents:

(3 * 3 * 3)^3 * ((-3)^2)^5 / 10

(9 * 3)^3 * ((-3)^2)^5 / 10

27^3 * ((-3)^2)^5 / 10

27^3 * 9^5 / 10

Continue evaluation exponents, breaking them down for simplicity:

27 * 27 * 27 * 9^5 / 10

27 * 27 * 27 * 9^5 / 10

729 * 27 * 9^5 / 10

19683 * 9^5 / 10

19683 * 9^2 * 9^2 * 9 / 10

19683 * 81 * 81 * 9 / 10

Then multiply the factors:

19683 * 81 * 729 / 10

19683 * 59049 / 10

1162261467 / 10

116226146.7

And there's your final answer.

You could also take advantage of the fact that X^N = (-X)^N if N is even by replacing -3 with 3 since 10 is even.

3^9 * (-3)^10 / 10

3^9 * 3^10 / 10

3^19 / 10

3 * 3^18 / 10

3 * (3^9)^2 / 10

3 * (3 * 3^8)^2 / 10

3 * (3 * (3^2)^4)^2 / 10

3 * (3 * ((3^2)^2)^2)^2 / 10

3 * (3 * (9^2)^2)^2 / 10

3 * (3 * 81^2)^2 / 10

3 * (3 * 6561)^2 / 10

3 * (19683)^2 / 10

3 * 387420489 / 10

1162261467 / 10

116226146.7