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

We are all aware of the concept of the number system in mathematics. There are infinite numbers spread over the number line. There are very large as well as very small numbers/ quantities in mathematics which cannot be clearly expressed as such. This is when the concept of exponents and powers comes into the picture.

Exponents and Powers

An exponent of a number represents the number of times the number has been multiplied by itself. Say, if 69 is multiplied n times by itself, then it would be depicted as:

69 x 69 x 69 x 69 x 69 x 69 x …….. x n = 69ⁿ.

Here, n is called the exponent of 69 and the expression 69ⁿ is read as 69 raised to the power n. Hence there is not much of a difference between the exponents and powers of the terms, since both of them represent the same notion.

Exponential Laws

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

p^m x pⁿ = p^m+n

• Division Law: When two exponents with the same bases but different powers are split, the base is increased to the difference between the two powers.

p^m ÷ pⁿ = p^m-n

• Negative Power Law: Any base that has a negative power, then it results in reciprocal but with 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 x q^m = (p x q)^m

• The power of power is multiplied by the former.

(p^m)ⁿ = p^mn

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, a^m × aⁿ = a^m+n]

= (5² × t⁻⁴)/(5⁻² × 2 × t⁻⁸)

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

= (5⁴ × t⁻⁴ + 8)/2

= 625t⁴/2

Similar Problems

Problem 1: Simplify: 1/2x⁻⁹⁹.

Solution:

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

1/ 2x^-99 = 

= x⁹⁹/2.

Problem 2: Simplify: 4/3x^-9.

Solution:

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

4/3x^-9 = 

Problem 3: Simplify: 12x⁹/5x⁶⁰.

Solution:

Using the property a^m/ aⁿ = a^{m – n}, which is known as the quotient law,

12x⁹/ 5x⁶⁰ = 

= 12x^-51/ 5

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

12x^-51/ 5 = 

Problem 4. Simplify: 3x²/10x⁵.

Solution:

Using the property a^m/ aⁿ = a^m-n, which is known as the quotient law,

3x²/10x⁵ = 

= 3x^-3/ 5

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

3x^-3/ 5 = 

Problem 5. Simplify 2x⁴/5y^-10.

Solution:

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

2x⁴/ 5y^-10 =