Simplify to positive exponents only
1. (X^4)^3
2. (X^-3)^-2
3. (4^-1)(3^-1)
4. (4ab^)(-5a^3b^2)
5. (5a^8b^-12)(-10a^3b^7)
just add exponents. If it winds up negative, it's in the denominator.
#3: (4^-1)(3^-1) = (4*3)^-1 = 12^-1 = 1/12
To simplify expressions with exponents, we can apply the rules of exponentiation. The rules are as follows:
1. (x^m)^n = x^(m * n): To raise a power to another power, we multiply the exponents.
2. x^(-m) = 1 / x^m: A negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent.
3. (x^m * y^n) = x^m * y^n: When multiplying terms with the same base, we can combine the exponents.
4. x^m / x^n = x^(m - n): When dividing terms with the same base, we can subtract the exponents.
Now let's simplify the given expressions:
1. (x^4)^3 = x^(4 * 3) = x^12
2. (x^-3)^-2 = 1 / (x^(-3 * -2)) = 1 / x^6 = x^(-6)
3. (4^-1)(3^-1) = (1/4)(1/3) = 1/12
4. (4ab^)(-5a^3b^2) = 4^(-5) * a^(-1 * 3) * b^(1 * 2) = 1 / (4^5 * a^3 * b^2)
5. (5a^8b^-12)(-10a^3b^7) = -50 * a^(8 + 3) * b^(-12 + 7) = -50a^11b^-5
Note that in the final expressions, we have simplified them to positive exponents only.