1. Quotient Properties Of Exponents Calculator
2. Quotient Properties

Exponents are used to show repeated multiplication. For example, 4³ means 4 · 4 · 4 = 64.

Quotient Properties Of Exponents Calculator

In this section, we will review basic rules of exponents.

Logarithm of a Quotient You can use the similarity between the properties of exponents and logarithms to find the property for the logarithm of a quotient. With exponents, to multiply two numbers with the. Exponent properties with quotients. Practice: Divide powers. Practice: Powers of products & quotients (structured practice) This is the currently selected item.

Product Rule of Exponents a^maⁿ = a^{m + n}

When multiplying exponential expressions that have the same base, add the exponents.

Example:

Multiply: 4x³ · −6x²

Solution:

Multiply coefficients: 4 · −6 = −24

Use the product rule to multiply variables : x³ · x² = x^{3 + 2} = x⁵

4x³ · −6x² = −24x⁵

Quotient Rule of Exponents

When dividing exponential expressions that have the same base, subtract the exponents.

Example:

Simplify:

Solution:

Divide coefficients: 8 ÷ 2 = 4

Use the quotient rule to divide variables :

Power Rule of Exponents (a^m)ⁿ = a^mn

When raising an exponential expression to a new power, multiply the exponents.

Quotient Properties

Example:

Simplify: (7a⁴b⁶)²

Solution:

Each factor within the parentheses should be raised to the 2^nd power:

(7a⁴b⁶)² = 7²(a⁴)²(b⁶)²

Simplify using the Power Rule of Exponents :

(7a⁴b⁶)² = 7²(a⁴)²(b⁶)² = 49a⁸b¹²

0 out of 0 correct.

• A surjective is a quotient map iff ( is closed in iff is closed in ).
• is a quotient map iff it is surjective, continuous and maps open saturated sets to open sets, where in is called saturated if it is the preimage of some set in .
• A quotient map does not have to be open or closed, a quotient map that is open does not have to be closed and vice versa.

Properties of quotient maps

• A restriction of a quotient map to a subdomain may not be a quotient map even if it is still surjective (and continuous).
  • If is a quotient map and is saturated with respect to , then if is open or closed, or is open or closed, then is a quotient map.
  • If is an open quotient map and is open, then is an open quotient map.
• The composite of two quotient maps is a quotient map.
• The product of two quotient maps may not be a quotient map.
  • If both quotient maps are open then the product is an open quotient map.
  • Another condition guaranteeing that the product is a quotient map is the local compactness (see Section 29).

Showing that a function is a quotient map

• If a continuous function has a continuous right inverse then it is a quotient map.
• A retraction is a quotient map. A retraction of onto is a continuous map such that for : .

Quotient topology and quotient space

• Let be a partition of the space with the quotient topology induced by where such that , then is called a quotient space of .
  • One can think of the quotient space as a formal way of 'gluing' different sets of points of the space.
• For to satisfy the -axiom we need all sets in to be closed.
  • For to be a Hausdorff space there are more complicated conditions.

Quotient spaces and continuous functions