Simplify 3x²(2xy – 3xy² + 4x²y³)

Last Updated : 22 Dec, 2023

The number system is a mathematical idea that we are all familiar with. On the number line, there are limitless numbers. In mathematics, there exist both huge and tiny numbers/quantities that cannot be explicitly represented as such. The idea of exponents and powers enters the picture at this point.

Exponents and Powers

The number of times a number has been multiplied by itself is represented by its exponent. For example, if 4 is multiplied by itself n times, the result is:

4 × 4 × 4 × 4 × 4 × 4 × …….. × n = 4ⁿ

The exponent of 2 is n, and the formula 2ⁿ is read as 2 raised to the power n. As a result, there is little difference between the exponents and powers of the words, because they both express the same concept.

Exponential Laws

Multiplication Law: The product of two exponents with the same base but different powers equals the base raised to the total of the two powers or integers, according to the exponent multiplication law.

p^m × pⁿ = p^m+n

Division Law: The base is raised to the difference between the two powers when two exponents with the same bases but different powers are divided.

p^m ÷ pⁿ = p^m-n

Negative Power Law: Negative Power Law states that if a base has a negative power, it will produce a reciprocal with a positive power or integer to the base.

p^-m = 1/p^m

Exponential Rules

According to this rule, if the power of any number is zero, the outcome will be unity or one.

p⁰ = 1

Different bases with equal powers in multiplication are multiplied together with the exponent put on the product.

p^m× q^m = (p × q)m

The power of power is multiplied by the former.

(p^m)ⁿ = p^mn

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

Similar Problems

Question 1. Simplify: $\frac{4ab^2(-5ab^3)}{10a^2b^2}$ .

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

Question 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 a^m/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:

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

= $\frac{p^{\frac{39}{14}}}{14}$

Question 3. 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

Question 4. Simplify: 3x²/10x⁵.

Solution:

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

3×2/10×5 = $\frac{3x^{2-5}}{10}$

= 3x^-3/ 5

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

3x-3/ 5 = $\frac{3}{10x^{3}}$ .

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