p^5 p^-4 z^2 divided by z^-5 zp^3
-21a^2 b^8 divided by -18a^-4 b^10
(2kr^3)^-3 divided by 4k^-2 r^6
y^2+3y+9/4
To simplify these expressions, we can apply the rules of exponents.
For the first expression:
1. Apply the power of a power rule: (p^5/p^-4)*(z^2/z^-5)*(zp^3).
- When we divide two powers with the same base, we subtract their exponents. So p^5/p^-4 becomes p^(5 - (-4)), which simplifies to p^9.
- Similarly, z^2/z^-5 becomes z^(2 - (-5)), which simplifies to z^7.
- Lastly, zp^3/1 is equivalent to zp^3.
2. Now combine the simplified terms: (p^9)(z^7)(zp^3).
- To multiply exponential terms with the same base, we add their exponents. So (p^9)(zp^3) becomes p^(9 + 1)zp^3, which simplifies to p^10zp^3.
3. The final expression is: p^10z^7p^3.
For the second expression:
1. Apply the power of a power rule: (-21a^2 b^8)/(-18a^-4 b^10).
- When we divide two powers with the same base, we subtract their exponents. So -21a^2/(-18a^-4) becomes -21a^(2 - (-4)), which simplifies to -21a^6.
- Similarly, b^8/b^10 becomes b^(8 - 10), which simplifies to b^-2.
2. Now combine the simplified terms: (-21a^6)(b^-2).
3. The final expression is: -21a^6/b^2.
For the third expression:
1. Apply the power of a power rule: (2kr^3)^-3/(4k^-2 r^6).
- When we raise a power to another power, we multiply the exponents. So (2kr^3)^-3 becomes 2^-3k^-3r^(3 * (-3)), which simplifies to 1/(8k^3 r^9).
- Similarly, (4k^-2 r^6) becomes 4k^-2 r^6.
2. Multiply the reciprocals: 1/(8k^3 r^9) * (1)/(4k^-2 r^6).
- When we multiply two fractions, we multiply the numerators and then multiply the denominators. So (1 * 1)/(8 * 4) is 1/32.
- For the variables, we subtract their exponents. k^3/k^-2 becomes k^(3 - (-2)), which simplifies to k^5.
- Similarly, r^9/r^6 becomes r^(9 - 6), which simplifies to r^3.
3. The final expression is: 1/32k^5r^3.