Simplify (p²-3p-40)/(2p³-24p²+64p)

Last Updated : 21 Dec, 2023

In mathematics, an expression composed of variables and constants as well as arithmetic operators like addition, subtraction, and so on is termed as an algebraic expression. Such equations are formed when the algebraic operators mentioned above are performed on any variable.

Types of Algebraic Expressions

Linear Expression: Such an algebraic expression where none of the exponents of the component variable is greater than unity. x − 2y is a linear expression because the exponents of both the variables x and y are 1.
Quadratic Expression: A quadratic expression is one that has the form ax² + bx + c, where a = 0. In other terms, a quadratic expression is an expression in which the greatest power of the variable is 2.
Cubic Expression: Here, the variables can have exponential values up to 3. It means that any such expression which has 3 as the highest exponent of the variable is a cubic expression.

Simplifying Algebraic Expressions

Simplification of an algebraic expression is the process of expressing an expression in the most efficient and compact way possible without changing the value of the original expression. Collecting similar phrases, which entails adding or removing terms from an expression, is a step in the process.

Steps to Simplify

Step 1: Factor the given equations(mostly in the case of quadratic and cubic expressions) and cancel out the common terms.

Step 2: Use the exponent rule to avoid grouping if the words contain exponents.

Step 3: Replace the terms that are similar with new ones.

Step 4: Add the values of the constants together.

Simplify (p²-3p-40)/(2p³-24p²+64p).

Solution:

Since the given expressions are quadratic and cubic, let’s factorize them first to see if any common terms can be eliminated.

$\frac{(p^2-3p-40)}{(2p^3-24p^2+64p)}\\=\frac{(p^2-8p+5p-40)}{2p(p^2-12p+32)}\\=\frac{p(p-8)+5(p-8)}{2p(p^2-8p-4p+32)}\\=\frac{(p+5)(p-8)}{2p(p-8)(p-4)}\\=\frac{p+5}{2p(p-4)}$ .

Sample Problems

Problem 1. Simplify: $\frac{p(p-8)+5(p-8)}{2p(p^2-8p-4p+32)}$ .

Solution:

Multiply the terms in the numerator, using the multiplication law of exponents.

$\frac{4ab^2(-5ab^3)}{10a^2b^2} = \frac{-20(a)^{1+1}(b)^{2+3}}{10a^2b^2}$

= $\frac{-2a^2b^5}{a^2b^2}$

Now apply the division law of exponents to evaluate.

= -2a^2-2b^5-2

= -2a⁰b³

= -2b³

Problem 2. Simplify: $\dfrac{(p^{1/7})^{49}}{\left(\dfrac{14p^{1/2}}{(p^{26})^{-1/7}}\right)}$ .

Solution:

Using the property (p^m)ⁿ = p^mn, we have:

$\dfrac{(p^{1/7})^{49}}{\left(\dfrac{14p^{1/2}}{(p^{26})^{-1/7}}\right)}=\dfrac{p^{49/7}}{\left(\dfrac{14p^{1/2}}{p^{-26/7}}\right)}$

Apply the property am/an = a^m-n in the denominator.

= $\dfrac{p^7}{{14p^{1/2-(-26/7)}}}$

= $\dfrac{p^7}{{14p^{59/14}}}$

Again applying the quotient law of exponents, we have:

$= \frac{{p^{7-\frac{59}{14}}}}{14}\\= \frac{p^{\frac{39}{14}}}{14}$

Problem 3. Simplify: 3x²(2xy – 3xy² + 4x²y³).

Solution:

P = 3x²(2xy − 3xy² + 4x²y³)

Using a^m.aⁿ = a^m+n, we have:

P = 6x²⁺¹y − 9x²⁺¹y² + 12x²⁺²y³

= 6x³y − 9x³y² + 12x⁴y³

Problem 4. Simplify: [25 × t^-4]/[5^-3 × 10 × t^-8].

Solution:

[25 x t^-4]/[5^-3 x 10 x t^-8] = (5² × t⁻⁴)/(5⁻³ × 5 × 2 × t⁻⁸ )

= (5² × t⁻⁴)/(5⁻³⁺¹ × 2 × t⁻⁸) [Since, am × an = am+n]

= (5² × t⁻⁴)/(5⁻² × 2 × t⁻⁸)

= (5²⁻⁽⁻²⁾ × t^{−4−(−8))}/2 [Since, am/an = am−n]

= (5⁴ × t^{−4 + 8})/2

= 625t⁴/2

Problem 5. Simplify: 1/2x^-99.

Solution:

Using the property a-m = 1/ am, which is known as the Negative exponent law,

1/ 2x^-99= $\frac{1}{2}x^{99}$

= x⁹⁹/2.

Problem 6. Simplify: 12x⁹/5x⁶⁰

Solution:

Using the property am/ an = am – n, which is known as the quotient law,

12x⁹/ 5x⁶⁰ = $\frac{12x^{9-60}}{5}$

= 12x^-51/ 5

Using the property a-m = 1/ am, which is known as the Negative exponent law,

12x^-51/ 5 = $\frac{12}{5x^{51}}.$