The quotient of power property states that when you have powers of the same base being divided, you can subtract the exponents of the denominator from the exponent of the numerator to determine the exponent of the quotient. This property is closely related to the product of power property, the power of a power property, and the power of a quotient property. The product of power property states that when multiplying powers with the same base, you can add the exponents, and the power of a power property states that when raising a power to a power, you can multiply the exponents. Finally, the power of a quotient property states that when raising a quotient to a power, you can raise both the numerator and denominator to that power.
Understanding the Structure of Quotient of Powers Property
The quotient of powers property states that when dividing powers of the same base, you can simply subtract the exponent of the divisor from the exponent of the dividend. In other words, for any real number base b and any non-zero real number exponents m and n:
$$\frac{b^m}{b^n} = b^{m-n}$$
This property holds true for any value of b, m, and n as long as b is not equal to 0.
How to Use the Quotient of Powers Property:
- Step 1: Identify the base (b) of both the dividend and divisor. Make sure the bases are the same.
- Step 2: Subtract the exponent of the divisor (n) from the exponent of the dividend (m) to get the new exponent (m-n).
- Step 3: Write the quotient as the base raised to the new exponent: b^(m-n).
Example:
(x^5) / (x^2) = x^(5-2) = x^3
Properties of the Quotient of Powers:
- Commutative Property: The order of the dividend and divisor does not affect the quotient. That is,
$$\frac{b^m}{b^n} = \frac{b^n}{b^m} = b^{m-n}$$
- Associative Property: The quotient of multiple powers with the same base can be grouped as desired. That is,
$$\frac{b^m}{(b^n)(b^p)} = \frac{b^m}{b^{n+p}} = b^{m-(n+p)}$$
- Distributive Property: The quotient of powers can be distributed over addition and subtraction. That is,
$$\frac{b^m \pm b^n}{b^p} = b^{m-p} \pm b^{n-p}$$
Table of Examples:
| Dividend | Divisor | Quotient |
|---|---|---|
| x^4 | x^2 | x^2 |
| 10^6 | 10^3 | 10^3 |
| (y^5)(y^2) | y^3 | y^4 |
| 2^7 / 2^3 | – | 2^4 |
| (a^m)(a^n) / a^p | – | a^(m+n-p) |
Question 1: What is the mathematical definition of the quotient of power property?
Answer: The quotient of power property states that when dividing powers with the same base, the exponent of the quotient is equal to the exponent of the dividend minus the exponent of the divisor.
Question 2: How can the quotient of power property be used to simplify expressions?
Answer: The quotient of power property can be used to simplify expressions by reducing the number of exponents and combining like terms. For instance, when dividing (x^5) by (x^2), the result can be simplified to x^(5-2) = x^3.
Question 3: What are the limitations of the quotient of power property?
Answer: The quotient of power property only applies when the bases of the powers are the same. If the bases are different, the property cannot be used to simplify the expression. Additionally, the property does not apply when raising a quotient to a power.
And that’s it for our quick dive into the quotient of powers property. I hope you found this explanation helpful. Remember, when you’re dealing with exponents, this property can come in handy for making calculations a lot easier.
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