# Simplify [(p^{3/4}q^{-2})^{1/3}]/[(p^{-2/3}q^{1/2})(pq)^{-1/3}]

Algebra is a discipline of mathematics concerned with the study of various symbols that represent quantities that do not have a fixed value or amount associated with them but instead vary or change over time in relation to some other factor. In the study of algebra, such symbols are referred to as variables, and the numbers associated with them are referred to as coefficients. They can be represented in a variety of ways, including forms and even English alphabets. In other words, algebra studies the representation of numbers using letters or symbols rather than the representation of their actual values.

**Expressions in Algebra**

In mathematics, an algebraic expression is a statement that is constructed utilizing variables and constants, as well as numerous arithmetic operations such as addition, subtraction, multiplication, division, exponential operations, root extraction such as square root, cube root, fourth root, and so on.

**Examples:**

- x + 22 is an algebraic expression in which x is the variable.
- x − y + 69 is an algebraic expression involving the variables x and y and the operation addition.
- 7x
^{2}+ 5xy + 44 is an algebraic expression involving the variables x and y and the operations addition, exponent, subtraction, and multiplication.

**Basic Terms**

**Variable:**In an algebraic expression, a variable is a phrase that can take on any value; its true value does not exist.**Coefficient:**It is a well-defined constant that is always used with a variable.**Operator:**Any mathematical operation such as addition, subtraction, multiplication, division, exponential operations, root extraction such as square root, cube root, fourth root, and so on and so forth is referred to as an operator.**Constant:**The constant is a word that is independent of both the coefficient and the variable and is well-defined in and of itself.

**Exponential Rules**

**Rule 1:** In multiplication, if two or more bases have the same powers, their powers are added together while keeping the base intact, i.e.,

a^{m}× a^{n}= a^{m+n}

**Rule 2:** If two or more bases in the division have the same powers, their powers are added together to keep the base intact. It should be noticed that the denominator’s power is subtracted from the numerator’s power, i.e.,

am ÷ an = a^{m-n}

**Rule 3:** Anything multiplied by a power of zero equals one.

a^{0}= 1

**Rule 4:** Power of a power is multiplied with the original one while keeping the base intact.

(a^{m})^{n}= a^{mn}

**Rule 5:** If two separate bases have the same power, multiply them and elevate the product to the power both b

a^{m}× b^{m}= (ab)^{m}

**Rule 6: **If a fractional exponent is specified, the numerator becomes the base’s power and the denominator becomes the root of the full expression, i.e.

a^{m/n}=

**Rule 7:** Reciprocate the base to make the power positive, i.e.

a^{-m }=

**Simplify **

**Solution:**

Apply the rule: (a

^{m }.b^{m})^{n}= a^{mn}b^{mn}=

=

Apply the rule: a

^{m}× a^{n = am+n in the denominator.}=

=

=

Apply the rule a

^{m}÷ a^{n}= a^{m-n}=

=

Apply the rule: a

^{-m}==

Hence, [(p=^{3/4}q^{-2})^{1/3}]/[(p^{-2/3}q^{1/2})(pq)^{-1/3}]p^{5/4}/q^{5/6}

**Similar Problems**

**Question 1. Simplify: (x ^{-7} y^{10x-8 y3)}**

**Solution:**

= (x

^{-7}x^{-8}) (y^{10}y^{3})Apply a

^{m}. a^{n}= a^{m+n}= x

^{-7-8}y^{10+3}= x

^{-15}y^{13}Apply a

^{-m}= 1/ a^{m}= y

^{13}/ x^{15}

**Question 2. Simplify: **

**Solution:**

= 3/5 (x

3) (y^{2}/x<^{3}/y^{2})= 3/5 (x

^{2-3}) (y^{3-2})= 3/5 (x

^{-1}) (y^{1})Apply a

^{-m}= 1/ a^{m}= 3y/5x

**Question 3. Simplify and write as positive exponents: (64x ^{-6} y^{6})^{5/6}.**

**Solution:**

Apply (abc)

^{m}= a^{m}b^{m}c^{m}(64x

^{-6}y^{6})^{5/6 }= 64^{5/6}x^{-30/6}y^{30/6}= 2

Apply a^{5}x^{-6}y^{5}^{-m}= 1/a^{m}Hence, (64x

^{-6}y^{6})^{5/6}= 32y^{5}/ x^{6}

**Question 4. Simplify and write as positive exponents: ****.**

**Solution:**

Since a

^{0}= 1.Thus, 4y

^{0}= 1We have: 2x

^{-10}/ 1 = 2x^{-10}Apply a

^{-m}= 1/a^{m}Hence, = 2/x

^{10}.

**Question 5. Simplify: ****.<**

**ong>**

**Solution:**

Apply (a

^{m})^{n}= a^{mn}, we have:[(2a

^{3})/ (3a^{5 × 3}]^{3 }= [(2a^{3})/ (3a^{15}]^{3}Using the property a

^{m}/a^{n}= a^{m-n}, we have:= 2/3[a

^{3-15}]^{3}= [2[a

^{-12}]^{3}]/3= 2a

^{-36}/3Hence, [(2a

^{3})/ (3a^{5})^{3}]^{3}=