How do I state which property of exponents justifies each step of this simplification?:
3^2x^6z^10/2^2y^18*x^4y^4z^4/5z^3*27z^6/2^2x^2y^3
Im really stuck on this problem, I've managed to figure out the first 5 problems but this one is troubling me.
To simplify the expression 3^(2x^6z^10) / (2^2y^18) * (x^4y^4z^4) / (5z^3) * (27z^6) / (2^2x^2y^3), let's analyze each step:
Step 1: Simplify the exponents within each factor.
- Using the property (a^m * a^n = a^(m+n)), multiply the exponents for each variable within the same factor.
- For example, 3^(2x^6z^10) can be expanded as (3^2) * (x^6) * (z^10), following the property of multiplying the exponents within the same base.
Step 2: Combine the common factors in the numerator and denominator.
- Combine the like terms of variables in the numerator and denominator by applying division of exponents.
- For example, x^6 / x^2 can be simplified as x^(6-2), following the property (a^m / a^n = a^(m-n)).
Step 3: Simplify the remaining fractions by simplifying the exponents.
- Evaluate the exponents and apply the property of exponentiation.
- For example, 2^2 = 2 * 2 = 4.
Step 4: Multiply the remaining factors together.
- Multiply the factors (numbers and variables) in the numerator and denominator.
By following these steps, you will simplify the given expression.
To simplify the given expression, we can apply several properties of exponents. Let's break down each step and state which property justifies it:
Step 1: Simplify within parentheses:
3^2 = 9
2^2 = 4
Step 2: Divide exponents of like bases (x^6 / x^4 = x^(6 - 4) = x^2):
x^6 / x^4 = x^(6 - 4) = x^2
Step 3: Multiply exponents of like bases (z^10 * z^4 = z^(10 + 4) = z^14):
z^10 * z^4 = z^(10 + 4) = z^14
Step 4: Divide exponents of like bases (z^14 / z^3 = z^(14 - 3) = z^11):
z^14 / z^3 = z^(14 - 3) = z^11
Step 5: Multiply exponents of like bases (z^11 * z^6 = z^(11 + 6) = z^17):
z^11 * z^6 = z^(11 + 6) = z^17
Step 6: Divide exponents of like bases (x^2 / x^2 = x^(2 - 2) = x^0):
x^2 / x^2 = x^(2 - 2) = x^0
Step 7: Any nonzero number raised to the power of 0 is equal to 1 (x^0 = 1):
x^0 = 1
Step 8: Divide exponents of like bases (y^18 / y^3 = y^(18 - 3) = y^15):
y^18 / y^3 = y^(18 - 3) = y^15
Step 9: Multiply exponents of like bases (y^15 * y^4 = y^(15 + 4) = y^19):
y^15 * y^4 = y^(15 + 4) = y^19
Step 10: Divide exponents of like bases (27 / 2^2 = 27 / 4 = 6.75):
27 / 2^2 = 27 / 4 = 6.75
Step 11: Divide exponents of like bases (27z^6 / z^0 = 27z^(6 - 0) = 27z^6):
27z^6 / z^0 = 27z^(6 - 0) = 27z^6
Step 12: Multiply exponents of like bases (6.75 * 27z^6 = 182.25z^6)
Step 13: Divide exponents of like bases (182.25z^6 / x^2 = 182.25z^6 / x^2)
Step 14: Divide exponents of like bases (182.25z^6 / y^19 = 182.25z^6 / y^19)
Therefore, the properties of exponents used in each step are:
- Step 2: Division of exponents when dividing like bases
- Step 3: Multiplication of exponents when multiplying like bases
- Step 4: Division of exponents when dividing like bases
- Step 5: Multiplication of exponents when multiplying like bases
- Step 6: Division of exponents when dividing like bases
- Step 8: Any nonzero number raised to the power of 0 equals 1
- Step 9: Multiplication of exponents when multiplying like bases
- Step 10: Division of exponents when dividing like bases
- Step 11: Division of exponents when dividing like bases.